Sufficent Conditions for the preservation of Path-Connectedness in an arbitrary metric space
Savvas Andronicou Emmanouil Milakis

	
	
	
	Sufficent Conditions for the preservation of Path-Connectedness in an arbitrary metric space
    S. Andronicou and E. Milakis
    
============================================================================================


		It is proven that if (X,d) is an arbitrary metric space and U is a path-connected subset of X with M:={x_i: i∈{1,2,…,k}}⊂ int(U), then the property of path-connectedness is also preserved in the resulting set U∖ M, provided that the boundary of each open ball of X is a non-empty and path-connected set. Moreover, under appropriate conditions we extend the above result in the case where the set M is countably infinite. As a consequence these results maintain path-connectedness for domains with holes.
		
	
	
	Mathematics Subject Classification: 54D05, 30L99
	
	Keywords: Path-connectedness, arbitrary metric spaces	
	 
	 

§ INTRODUCTION

	The assumptions of connectedness and path-connectedness play a critical role when studying questions in Analysis, Topology and Partial Differential Equations. These conditions are present in a variety of theorems and applications, from regularity in free boundary problems, stability questions in fractured domains and derivation of interpolation inequalities to global bifurcation theory and the structure of solutions sets to vector equilibrium problems. For instance in the study of free boundaries in variational inequalities one seeks regularity of domains with finite perimeter in the sense of Caccioppoli and De-Giorgi (<cit.>, <cit.>) while these domains verify the condition that the interior of its complement is composed by finite many connected components (see <cit.>). In another occasion <cit.>, the authors obtain geometric properties of nodal domains of solutions to certain uniformly elliptic equations. Again, the assumption of path-connectedness to these nodal domains leads to major impeccable results for instance when trying to obtain uniform bounds on frequency functions for solutions with same zero set. In general, there is a vast bibliography in these topics but we could, at least and additionally to the above, refer to  <cit.>, <cit.>, <cit.>, <cit.>, <cit.> and <cit.>.
	
	 Having in mind the above mentioned topics, one ends up with an abstract question regarding path-connectedness.  
	 	 
	 Question: Assume that (X,d) is an arbitrary metric space and U is a path-connected subset of X. If one removes a subset M from U, is it true that resulting set U∖ M becomes also path-connected? 
	 
	 Although this question seems to have a fair answer for Euclidean settings, the case of arbitrary metric space appears a bit more tricky. To the best of our knowledge, the question remains open even in the case when M is finite or countably infinite set of points. The main purpose of the present article is to fill exactly this gap, giving the proofs in full details. We prove that if (X,d) is an arbitrary metric space and U is a path-connected subset of X with M:={x_i: i∈{1,2,…,k}}⊂ int(U), then the resulting set U∖ M is also path-connected,  provided that the boundary of each open ball of X is a non-empty and path-connected set. Moreover, under appropriate conditions, the above result holds true in the case where the set M is countably infinite. Although this question has an abstract nature and is of independent interest across several topics of Mathematical Analysis, the motivation of the present paper is a bit more focused. In a forthcoming article <cit.>, we are establishing properties of nodal domains for solutions to general elliptic type equations that include corkscrew conditions, Carleson type estimates and boundary Harnack inequalities. These estimates play a crucial role, for instance, in the study of two phase free boundary problems in which the positive and negative parts of a solution satisfy two (perhaps different) elliptic equations and the free boundary condition  involves normal derivatives from positive and negative sides. The proofs follow the fundamental ideas of <cit.> and when one tries to reproduce the proof of Theorem 5.1 of <cit.> the corresponding elements in the partition of the nodal domains need to stay path-connected, a condition which leads to the exact question we have imposed above. 
	 
	
	 
	 The structure of the paper is as follows. In Section <ref> we provide the appropriate definitions, notations and present the main results while Section <ref> is devoted to the proofs of the main theorems.  
	
	
	
	

§ DEFINITIONS AND MAIN RESULTS

	
	
	

 §.§ Definitions

	In this section we give some necessary definitions and results for connected and path-connected metric spaces (for more details see <cit.>, <cit.>, <cit.>).
	 
	A metric space[ In this paper, when we refer to the notion of a metric space (X,d), we mean a pair of objects (X,d), where the underlying set X is an an arbitrary set (it could also be the empty set) and d is a map that fulfills the axioms of a metric. In bibliography, some authors exclude the empty metric space. We insist on using the more general definition. ] ( X,d) is called connected if there is no open partition i.e there is no pair of sets {A,B} such that the following holds:
		
    (i) X=A∪ B, (ii)  A,B≠∅, (iii)  A∩ B=∅, (iv) A, B are open.

		
	
		A metric space ( X,d) is called path-connected if for every x,y∈ X, x≠ y there exists a continuous map γ:I⊂→ X where I is an interval, such that x,y∈γ(I).
	
To be more specific, when we refer that a genuine subset A of X fulfills the property of connectedness (path-connectedness), we mean that the metric subspace ( A,d_A) is connected (path-connected). We see that the notion of path-connectedness is stronger than then notion of connectedness. For example, the topologist's sine curve, i.e the set {(0,0)}∪ Gr(f) where f:(0,1]→, f(x):=sin(1/x). In specific this set is connected in the eucledian space, but it is not path-connected, since there is no path connecting the origin to any other point on the space.

A well known result that relates continuity and connectedness is the following:

	Let (X,d ), (Y,ρ) metric spaces, and f:X→ Y be a continuous function. If (X,d ) is connected space, then the image f(X) is connected subset of Y.

		Next we present our main results.

	

 §.§ Main Results

	 Let (X,d) be a metric space and U a path-connected subset of X. We define M:={x_1,x_2,… x_k}⊂ int(U). If for every x∈ X and δ>0, the boundary of the open ball ∂ B_d(x,δ) is a non-empty and path-connected set, then the set U∖ M is also path-connected.
	
In the case of violation of the assumption required in the above theorem, regarding the boundary of every open ball in the metric space to be path-connected, can ultimately lead to a violation of the result of the theorem. A trivial example that imposes the necessity of the assumption, is for the usual metric space  ( ,ρ_·). In specific, in this space the boundary of every open ball B_ρ_·( x,r)=( x-r,x+r) is the set {x-r,x+r}. Setting, now  U to be any bounded interval in , (therefore U is path-connected) then for any  internal point x_0 of U, we conclude that the set U∖{x_0} is not path-connected, since is not even connected set. Moreover, even if the condition of path-connectedness of the boundary of every open ball holds, the violation of M being a subset of the interior of U, can ultimately lead to a violation of the result of the theorem. An example that illustrates the last case, is the metric space (^2,d_2) where in this space the boundary of every open ball is path-connected. Now, let y_0 be any fixed real number and let A  be any finite subset of . Then, we set as U to be U:={(x,y_0): x∈} and M:={(x,y_0): x∈ A}. We observe that M⊂ U but int(U)=∅. Therefore M⊈ U. Easily we conclude that U is path-connected but U∖ M is not path-connected.

Next, we present a theorem that provides sufficent conditions for preservation of connectedness of a metric space after the extraction of a countably infinite subset. 


	Let (X,d) be a metric space, U a path-connected subset of X with M⊂ U. We call that the set U satisfies the property (U-M) if and only if:

	∀ x,y∈ U∖ M, ∀x̃∈ M, if γ:I→ U is a path that connects the points x,y and (x_j)_j∈ a sequence of discrete points of M such that:
	
    ∃ z∈{x_j: j∈}∖{x̃}, dist( x̃,{x_j: j∈}∖{x̃})  =d(x̃,z)

then there exists continuous map γ^*_z,x̃:[t_z,t_x̃]→ U such that z=γ^*_z,x̃(t_z), x̃=γ^*_z,x̃(t_x̃) and γ^*_z,x̃((t_z,t_x̃) ) ∩ M=∅. 


	Let (X,d) be a metric space, U a path-connected subset of X with M⊂ int(U) closed  countably infinite such that iso(M)=M. If (U-M) property holds and for every x∈ X and δ>0, ∂ B_d(x,δ) is a non-empty and path-connected set, then the set U∖ M is also path-connected.



 











	
	
	
	

§ NOTATIONS

	Let (X,d) be a metric space and A⊂ X.

	B_d(x,δ):={z∈ X : d(z,x)<δ} the usual open ball with center x∈ X and radious δ>0 

	int(A):={x∈ X: ∃δ>0, B_d(x,δ)⊂ A}, the internal of the set A

	ext(A):=int(X∖ A), the external of the set A

	Cl(A)≡A̅:={x∈ X: x limit point of the set A}, the closure of the set A

	∂ A:=A̅∩ Cl( X∖ A), the boundary of the set A

	iso(A):={x∈ X: ∃δ_x>0, B_d(x,δ_x)∩ A={x}}, the set of isolated points of the set A

	For a continuous function γ:I→ X, where I is an interval in ,  we use the following symbolism:
	
		
  * γ(I):={γ(t)|  t∈ I} the trace of the curve γ on the interval I.
		
  * For x,y∈γ(I), let x=γ(t_x), y=γ(t_y) and without loss of generality, let t_x<t_y, then we set: 
		γ_xy:={γ(t)| t∈[t_x,t_y]}, i.e the part of curve γ that connects the points x and y.
	

	
	
	

§ PROOF OF MAIN RESULTS

	The proof of the above theorems is implemented through the following fundamental to our analysis proposition:
 
	
		Let (X,d) be an arbitrary metric space, U⊂ X, x_0∈ int(U), z∈ X∖{x_0}, such that ∀δ>0, ∂ B_d(x_0,δ)≠∅. If γ:I→ X is a continuous function defined on the interval I⊂ such that x_0=γ(t_x_0),z=γ(t_z), where t_x_0,t_z∈ I, then there exists δ_0>0 such that
		
    ∅≠γ(Ĩ) ∩∂ B_d(x_0,δ)⊂ U, ∀δ∈( 0,δ_0]

	where Ĩ:=
		[t_x_0,t_z],    t_x_0<t_z

		[t_z,t_x_0],    t_z<t_x_0.
	
		

			
    < g r a p h i c s >

		

	
		Without loss of generality, let t_x_0<t_z, i.e Ĩ:=[t_x_0,t_z]. From the assumption that x_0∈ int(U), there exists ϵ_x_0>0 such that B_d(x_0,ϵ_x_0)⊂ U. We define 
		
    δ_0:=1/2min{ϵ_x_0,d(z,x_0)}.

	We observe that δ_0>0, since x_0≠ z. We will prove that ∀δ∈(0,δ_0], γ(Ĩ)∩∂ B_d(x_0,δ)≠∅.

	Let suppose the opposite of what we seek to prove i.e 
	
    ∃δ̃∈(0,δ_0], γ(I)∩∂ B_d(x_0,δ̃)=∅ .

Claim z∈ ext( B_d(x_0,δ̃) ) ≡ int( X∖ B_d(x_0,δ̃)).

In specific we prove that B_d(z,δ̃)⊂ X∖ B_d(x_0,δ̃). Indeed, let y∈ B_d(z,δ̃), equivalently, d(y,z)<δ̃. For what we are asking to prove, it is enough to show that y∉ B_d(x_0,δ̃), equivalently d(y,x_0)≥δ̃. Let suppose that the last does not hold i.e d(y,x_0)<δ̃. Then,

    d(x_0,z)≤ d(x_0,y)+d(y,z)<2δ̃≤ 2δ_0≤ 21/2d(x_0,z)=d(x_0,z).

So we obtain d(x_0,z)<d(x_0,z) which is a contradiction.

We continue now with the fact that the metric space X receives a partition  in terms of the open ball B_d(x_0,δ̃)  ( i.e int( B_d(x_0,δ̃))=B_d(x_0,δ̃)):

    X=B_d(x_0,δ̃) ∪∂ B_d(x_0,δ̃)∪ ext( B_d(x_0,δ̃)) .

As a result from the above, we obtain

    γ(Ĩ)   =γ(Ĩ)∩ X 
       =(γ(Ĩ)∩ B_d(x_0,δ̃) ) ∪(γ(Ĩ)∩∂ B_d(x_0,δ̃)  ) ∪( γ(Ĩ)∩ ext( B_d(x_0,δ̃) ))
       =^(<ref>)(γ(Ĩ)∩ B_d(x_0,δ̃) )∪( γ(Ĩ)∩ ext( B_d(x_0,δ̃) ))
       =V_1∪ V_2

where V_1:=γ(Ĩ)∩ B_d(x_0,δ̃) and V_2:=γ(Ĩ)∩ ext( B_d(x_0,δ̃)) .

First we see that from the Principle of Inheritance, the sets V_1, V_2 are open in the metric subspace (γ(Ĩ),d_γ(Ĩ)). Moreover, V_1≠∅, since x_0∈γ(Ĩ)∩ B_d(x_0,δ̃) and V_2≠∅, since from the previous claim, we have z∈ ext(B_d(x_0,δ̃) ) and z∈γ(Ĩ). Clearly, we see that V_1∩ V_2=∅, since 

    B_d(x_0,δ̃)∩ ext( B_d(x_0,δ))= B_d(x_0,δ̃)∩ int(X∖ B_d(x_0,δ))⊂ B_d(x_0,δ̃)∩(X∖ B_d(x_0,δ))=∅.

Consequently, the pair of sets {V_1, V_2} is an open (with respect to the relative metric d_γ(Ĩ)) partition of γ(Ĩ). From the last we conclude that the set γ(Ĩ) is not connected. The latter is contradicted with the fact that, since Ĩ is an interval in , therefore a connected set and the function  γ:Ĩ→ X, is continuous, it follows from Theorem <ref> that γ(Ĩ) will be connected. As a result from the above analysis, we receive that:  
∀δ∈(0,δ_0], γ(Ĩ)∩∂ B_d(x_0,δ)≠∅.

 Moreover, since 0<δ≤δ_0≤1/2ϵ_x_0<ϵ_x_0, it follows that:

    γ(Ĩ)∩∂ B_d(x_0,δ)   ⊂γ(Ĩ)∩B̅_d(x_0,δ)
       ⊂γ(Ĩ)∩ B_d(x_0,ϵ_x_0)
       ⊂ ^( B_d(x_0,ϵ_x_0)⊂ U) γ(Ĩ)∩ U⊂ U.


 	
	It is worth noting that, in case where the metric space (X,d)  has the property: ∀ x∈ X, ∀δ>0, ∂ B_d(x,δ) is path-connected, then automatically (by the definition of path-connectedness) it is extracted that ∂ B_d(x,δ)≠∅, ∀ x∈ X, ∀δ>0. Moreover, from the above proposition, an interesting result is derived regarding the minimum number of intersection points of the curve with the boundary of the ball.
	
	
		Let (X,d) is an arbitrary metric space, U⊂ X,   x_0∈ int(U), x,y∈ X∖{x_0} with x≠ y and ∀δ>0, ∂ B_d(x_0,δ)≠∅. If γ:I→ X is a continuous function defined on the interval I⊂, such that x,y,x_0∈γ(I) and γ_xx_0∩γ_x_0y={x_0}, then there exists δ̃_̃0̃>0 such that 
		
    card( γ(I)∩∂ B_d( x_0,δ) )≥ 2, ∀δ∈( 0,δ̃_̃0̃] .

	
	
		Let x=γ(t_x), y=γ(t_y), x_0=γ(t_x_0). Since x≠ y, it follows that t_x≠ t_y. Without loss of generality, let us suppose that t_x<t_y. 

		Claim t_x<t_x_0<t_y

		Since x,y∈ X∖{x_0}, we obtain that t_x≠ t_x_0 and t_y≠ t_x_0. Let us suppose that  t_x<t_y<t_x_0. We see that
		
    γ_yx_0:={γ(t)| t∈ [t_y,t_x_0]}⊂{γ(t)| t∈ [t_x_0,t_x]}:=γ_x_0x

	and from the above inclusion we also receive that {x_0,y}⊊γ_yx_0∩γ_x_0x, which contradicts with the assumption that γ_yx_0∩γ_x_0x={x_0}. We arrive in a similar way at a contradiction for the case t_x_0<t_x<t_y. Consequently the above claim is true.

	Now, since x_0∈ int(U), it exists ϵ_x_0>0, such that B_d(x_0,ϵ_x_0) ⊂ U. Next we define
	
    δ̃_0:=1/2min{ϵ_x_0,d(x,x_0),d(y,x_0)}.

It is clear that δ̃_0>0 and 

    δ̃_0≤δ^(1)_0:=1/2min{ϵ_x_0,d(x,x_0)} andδ̃_0≤δ^(1)_0:=1/2min{ϵ_x_0,d(x,x_0)}.

Then from the above relations (<ref>), combined with the Proposition <ref> we conclude that ∀δ∈(0,δ̃_0],

    K_1^(δ):=γ_xx_0∩∂ B_d(x_0,δ)≠∅ and 
    	K_2^(δ):=γ_x_0y∩∂ B_d(x_0,δ)≠∅

	where γ_xx_0:={γ(t)| t∈ [t_x,t_x_0]} and γ_x_0y:={γ(t)| t∈ [t_x_0,t_y]}. Let z_1^(δ)∈ K_1^(δ),z_2^(δ)∈ K_2^(δ). It holds that K_1^(δ)∩ K_2^(δ):=∅. Indeed, if we supose that there exists z_δ∈ K_1^(δ)∩ K_2^(δ), then z_δ∈γ_xx_0∩γ_x_0y. By the assumption in the corollary, we conclude that z_δ=x_0. Therefore, since z_δ∈∂ B_d(x_0,δ) we receive that x_0∈∂ B_d(x_0,δ), which is a contradiction in an arbitrary metric space.[Let (X,d) be a metric space, x_0∈ X and δ>0. Then B_d(x_0,δ)∩∂ B_d(x_0,δ)=∅. Let us suppose the opposite of what we seek to prove. I.e let z∈ B_d(x_0,δ)∩∂ B_d(x_0,δ). We define ϵ_z:=δ-d(z,x_0). Notice that ϵ_z>0 since z∈ B_d(x_0,δ). Then, we see that B_d(z,ϵ_z)∩(X∖ B_d(x_0,δ) )=∅, since B_d(z,ϵ_z)⊂ B_d(x_0,δ). This violates the definition of z being a limit point of X∖ B_d(x_0,δ). Therefore, z∉ Cl( X∖ B_d(x_0,δ) ). Consequently z∉∂ B_d(x_0,δ), which contradicts the initial assumption for z.] Therefore, K_1^(δ)∩ K_2^(δ)=∅ resulting that z_1^(δ)≠ z_2^δ. In conclusion, z_1^(δ),z_2^(δ)∈γ(I)∩∂ B_d(x_0,δ), therefore card( γ(I)∩∂ B_d( x_0,δ) )≥ 2, ∀δ∈( 0,δ̃_0].

	
	
	
	
	
	
	

 §.§ Proof of Theorem <ref>

	
			We begin the proof of the main result for the case k=1.
		Case: (k=1 )

		In this case we prove that the set U∖{x_0} is path-connected. Let x, y∈ U∖{x_0} two randomly selected points such that x≠ y. From the assumption of the theorem, U is path-connected, therefore there exists a continuous function γ:I→ U, where I is an interval, such that x=γ(t_x) and y=γ(t_y), with t_x,t_y∈ I.  Since x≠ y it follows that t_x≠ t_y. Without loss of generality, we consider t_x<t_y. We distinguish the following cases:
		
			
  (i) x_0∉γ([t_x,t_y])

			Therefore, γ([t_x,t_y])⊂ U∖{x_0}.
			
  (ii) x_0∈γ([t_x,t_y])

			Let t_x_0∈(t_x,t_y) such that x_0=γ(t_x_0). We define:
			
    J:={t∈[t_x,t_y]| γ(t)=x_0}, t̃:=inf J, t^*:=sup J.

			We observe that J≠∅, since t_x_0∈ J. Moreover, J is lower and upper bounded by t_x and t_y respectively. Therefore, t̃, t^* are well defined. In specific, t_x≤t̃≤ t^*≤ t_y. From the definition of  t̃, t^* we receive that both of them are limit points of J. Then by the sequential description of limit points, there are sequences (l_n)_n∈,(c_n)_n∈⊂ J such that l_nt̃ and c_nt^*. Using now the continuity of γ, we receive that γ(l_n)γ(t̃) and γ(c_n)γ(t^*). Since γ(l_n)=γ(c_n)=x_0, ∀ n∈ by the uniqueness of limits for sequences, we finally obtain that γ(t̃)=x_0=γ(t^*).

			Next we claim that t̃>t_x, since if t̃=t_x, then γ(t̃)=γ(t_x), i.e x_0=x, which contradicts with the assumption of the theorem. Similarly, we prove that t^*<t_y. Summarizing, we have that t_x<t̃≤ t^*<t_y.

			From the assumption x_0∈ int(U), there exists ϵ_x_0>0 such that B_d(x_0,ϵ_x_0)⊂ U. Next we define 
			
    δ_0=1/2min{ϵ_x_0,d(x_0,x), d(x_0,y)}.

			Clearly we see that 
			
    δ_0≤δ_0^(1):=1/2min{ϵ_x_0,d(x_0,x)} and δ_0≤δ_0^(2):=1/2min{ϵ_x_0,d(x_0,y)}.

			Using now the Proposition <ref> we obtain that ∀δ∈(0,δ_0],
			
    ∅≠ K_1≡γ([t_x,t̃])∩∂ B_d(x_0,δ)⊂ U and∅≠ K_2≡γ([t^*,t_y])∩∂ B_d(x_0,δ)⊂ U.

			Let z_1∈ K_1 and z_2∈ K_2. As a result from the last, 
			
    ∃ t_z_1∈ [t_x,t̃ ], γ(t_z_1)=z_1∈∂ B_d(x_0,δ) and ∃ t_z_2∈ [t^*,t_y], γ(t_z_2)=z_2∈∂ B_d(x_0,δ).

			Initially, we claim that t_z_1<t̃≤ t^*<t_z_2. Indeed, let t_z_1=t̃. Then, γ(t_z_1)=γ(t̃), i.e z_1=x_0 resulting that x_0∈∂ B_d(x_0,δ), which is a contradiction. Consequently, t_z_1<t̃. Similarly, we prove that  t^*<t_z_2.

			Next, using the assumption of theorem, the boundary of ball, ∂ B_d(x_0,δ) is path-connected set. Therefore, there exists a continuous function γ̃:[t_z_1,t_z_2]→∂ B_d(x_0,δ) such that γ̃(t_z_1) =z_1 and γ̃(t_z_2) =z_2. We notice that, 
			
    γ̃([t_z_1,t_z_2])⊂∂ B_d(x_0,δ)⊂ U∖{x_0}.

			We define the following function γ^*:[t_x,t_y]→ U with
			
    γ^*(t):=
    				γ(t)      t ∈ [t_x,t_z_1]
    γ̃(t)    t∈[t_z_1,t_z_2]
    γ(t)     t∈[t_z_2,t_y].

			Spefically, the function γ^* is continuous and γ^*([t_x,t_y])⊂ U∖{x_0}. Indeed, in order to prove the last, it is enough to valid the following:
			
    γ^*([t_x,t_z_1]∪[t_z_2,t_y] ) ⊂ U∖{x_0}.

			Let us suppose that there exists t̅∈[t_x,t_z_1] such that γ^*(t̅)=γ(t̅) =x_0. Then it follows that t̅∈ J. Therefore, t̃:=inf J≤t̅ which is a contradition, since t̅≤ t_z_1<t̃. In a similar way, if we suppose the existence of an element t̅∈[t_z_2,t_y] such that γ^*(t̅)=γ(t̅) =x_0, it follows that t̅∈ J. From the last we derive that t^*:=sup J≥t̅ which is a contradiction since t̅≥ t_z_2>t^*. Consequently, the inclusion (<ref>) is true.
		
		From the above two cases, we proved that for any two points x,y∈ U∖{x_0} with x≠ y there exists a continuous function γ:I⊂→ U∖{x_0}, where I is an interval, such that x,y∈γ(I). In consequence, the set U∖{x_0} is path-connected.

		We proceed now to the general case where k>1.
		Case: (k>1)

		First, we define inductively the following finite sequence of sets (V_i)_i∈{1,2,…,k}:
		
    V_1:=U∖{x_1},  V_i:= V_i-1∖{x_i}, i=2,…,k.

		From the previous case k=1, since U is path-connected set and x_1∈ int(U), we receive that V_1 is path-connected set. Let us suppose that V_i is path-connected for some random index i∈{1,k-1}. Then, we notice that from the definition of V_i+1,  the following holds: V_i+1:=V_i∖{x_i+1}, where V_i is path-connected set (from the inductive step) and x_i+1∈ int(V_i). Using now the previous case, we conclude that V_i+1 is path-connected set. From the last finite inductive scheem, we receive that ∀ i∈{1,2,…,k}, V_i is path-connected set. Specifically, the set V_k=U∖{x_1,x_2,… x_k} is path-connected.
	

	
	
	
	Next we proceed to the proof of the Theorem <ref>.
	

 §.§ Proof of Theorem <ref>

	
		Let M be a closed, countable infinite subset of int(U) with all of its points to be isolated i.e iso(M)=M. We proceed to the proof that U∖ M is path-connected. Let x,y∈ U∖ M two randomly selected points such that x≠ y. From the assumption of the theorem, U is path-connected, therefore there exists a continuous function γ:I→ U, where I is an interval such that x=γ(t_x) and y=γ(t_y) for some t_x,t_y∈ I. Since x≠ y it follows that t_x≠ t_y. Without loss of generality, we consider t_x<t_y. Initially we set S:=M∩γ([t_x,t_y]). We distinguish the following cases:

		Case: ( S=∅)

			Therefore, we obtain γ([t_x,t_y])⊂ U∖ M.

		Case: (S≠∅ )

			Initially, we define
			 
    t^*:=sup{t∈[t_x,t_y]: γ(t)∈ M}.

		 
		 	The real number t^* is well defined and also satisfies the following:
		 	
		 		
  (i) γ(t^*)∈ S∖{x,y}, equivalently γ(t^*)∈ M∩γ((t_x,t_y ) )
		 		
  (ii) t_x<t^*<t_y
		 		
  (iii) γ( (t^*,t_y)) ∩ M=∅.
		 	
		 
	 (of Claim <ref>)
	 	First we prove that, t^* is well defined. Indeed, from the assumption of theorem, S≠∅ i.e there exists z∈ S, equivalently, there exists z∈ M, such that z=γ(t_z) for t_z∈[t_x,t_y]. From the last, we conclude that z∈ J:={ t∈[t_x,t_y]: γ(t)∈ M }. Therefore, J≠∅. Obviously, J is upper bounded from t_y  therefore t^* is well defined. Next, we proceed to the proof of (i)-(iii).
	 	
	 		
  (i) Since t^*=sup J is a limit point of J there exists (t̃_l)_l∈⊂ J such that t̃_lt^*. By the continuity of γ, it follows that γ(t̃_l)γ(t^*).  Since (t̃_l)_l∈⊂ J we obtain that, γ(t̃_l)∈ M. Therefore γ(t^*) is a limit point of M. By the assumption M is closed, and as a result γ(t^*) ∈ M. Moreover, since  ∀∈, t̃_l∈ J⊂ [t_x,t_y], it follows that ( γ(t̃_l)) ⊂γ([t_x,t_y] ) . From the last, we conclude also that γ(t^*) is limit point of γ([t_x,t_y] ) . But the image γ([t_x,t_y] ) is closed as a continuous image of the compact set [t_x,t_y]. As a result we receive finally that γ(t^*)∈γ([t_x,t_y] ). Now, if γ(t^*)=x=γ(t_x) then, due to the fact that γ(t^*)∈ M, it follows then x∈ M, where the last contradicts with the assumption that x∉ M. By a similar argument, we prove that γ(t^*)≠γ(t_y)=y. In conclusion, we receive that γ(t^*)∈ M∩γ(( t_x,t_y)  ).
	 		
  (ii) First we see that t_x∈ J and from the definition of t^* we obtain that t_x≤ t^*. Furthemore, t_y is an upper bound of J, therefore t^*≤ t_y. If now t_x=t^* then x=γ(t_x)=γ(t^*) where from (i) we obtain that γ(t^*) ∈ M, i.e x∈ M. The last contradicts with our initial assumption, that x∈ U∖ M. As a result t_x<t^*. With a similar argument, we prove that t^*<t_y.
	 		
  (iii) Let us suppose the opposite of what we week to prove i.e there exists z∈ M such that z=γ(t_z) with t_z∈ (t^*,t_y)⊂^(ii)[t_x,t_y].  It is obvious that t_z∈ J. Therefore, t_z≤sup J=t^* which is a contradiction, since t_z>t^*.
	 	
	 
 Next we define inductively the following sequences (t_k)_k∈, (t'_k)_k∈ of points in the interval [t_x,t_y].
 
    We set  t_0:=t_x, t'_0:=t_x. For k∈ we define:
    
     		t_k:=inf{ t∈[t'_k-1,t^*] : γ(t)∈ M∖( ⋃_i=0^k-1{γ(t'_i) }) } ,    if  γ(t'_k-1)≠γ(t^*) 
    
     			t^*,    if γ(t'_k-1)= γ(t^*)
     		
     t'_k:=sup{ t∈[t'_k-1,t^*]: γ(t)=γ(t_k) }.

 
[Properties of the sequences (t_k)_k∈, (t'_k)_k∈]
		The sequences (t_k)_k∈, (t'_k)_k∈ have the following properties:
		
			
  (i) They are well defined.
			
  (ii) ∀ k∈, γ(t_k)=γ(t'_k)∈ M.
			
  (iii) ∀ k∈, t_x<t_k≤ t'_k≤ t^*.
			
  (iv) If there exists k_0∈ such that t_k_0=t^*, (or t'_k_0=t^*) then ∀ k≥ k_0, t_k=t^*=t'_k,
 (resp. ∀ k≥ k_0+1, t_k=t^*=t'_k ). I.e in this case the sequences  (t_k)_k∈, (t'_k)_k∈ are finally constant.
			
  (v) Let J:={ k∈: γ(t'_k) =γ(t^*) }. 
 We define k^*:=
				inf J,    if J≠∅

				+∞,    if J=∅ and  K:={1,2,…,k^*},    if k^*∈
,    if k^*=∞.
 Then, the following hold:
				
					
  (a)t_k-1<t^*, and  t'_k-1<t^*,  ∀ k∈ K
					
  (b) γ(t_k-1)=γ(t'_k-1) ≠γ(t^*), ∀ k∈ K
					
  (c) t'_k-1<t_k,  ∀ k∈ K
					
  (d) t_k-1<t_k,   ∀ k∈ K
					
  (e) t'_k-1<t'_k,  ∀ k∈ K
				 
			
  (vi) γ(t_i)≠γ(t_j), ∀ i,j∈ K, i≠ j
			
  (vii) For all k∈ K, it holds that: γ((t'_k-1,t_k)  )∩ M=∅.
			 

		 
		 (of Claim <ref>)
		 	
		 		
  (i) We will prove by induction that the sequences (t_k)_k∈, (t'_k)_k∈ are well defined. First, for k=1, we observe from the Claim <ref> (i), that γ(t'_0)=γ(t_x)=x≠γ(t^*). Therefore, t_1=inf J_1 where,  J_1={ t∈[t_x,t^*]: γ(t)∈ M∖{x}}. Initially, we notice that J_1≠∅, since t^*∈ J_1 (see Claim <ref> ) (i). Furtheremore, J_1 is lower bounded from t_x. Therefore t_1=inf J_1 is well defined. Moreover, t'_1:=sup J'_1 where J'_1:={ t∈[t_x,t^*]: γ(t)=γ(t_1)}. It holds that t^* is an upper bound of J'_1 and also non empty set, since t_1∈ J'_1. Indeed, from the fact that t_x is a lower bound of J_1, it follows that t_x≤ inf J_1 =t_1 and since t^* is an upper bound of J_1 we obtain that t_1≤ t^* . In consequence, t_1∈[t_x,t^*].

		 		Let us suppose now that t_k and t'_k are well defined for random k∈. We will prove  that t_k+1 and t'_k+1  are well defined as well. We distinguish the following cases for γ(t'_k).
		 		
		 			
  * Let γ(t'_k)=γ(t^*). Then by the definition of t_k+1, we receive that t_k+1=t^*. Moreover, t'_k+1=sup J'_k+1 where J'_k+1:={ t∈[t'_k,t^*]: γ(t)=γ(t_k+1)}={ t∈[t'_k,t^*]: γ(t)=γ(t^*)}. It is clear that J'_2≠∅, since t^*∈ J'_k+1 and also J'_k+1 is upper bounded from again from t^*. In consequence, t'_k+1 is well defined.
 		 			
  * Let γ(t'_k)≠γ(t^*). Then by the definition of t_k+1, we receive that t_k+1=inf J_k+1, where J_k+1:={ t∈[t'_k,t^*]: γ(t)∈ M∖( ∪_i=0^k{γ(t'_i)})  }.
		 			Initially, we observe that J_k+1 is lower bounded from t'_k. Furtheromore, we claim that J_k+1≠∅. In specific, we claim that t^*∈ J_k+1. First, we notice from Claim <ref> (i), that γ(t^*)∈ M. Moreover, from the assumption, we receive that γ(t^*)≠γ(t'_k) and γ(t'_0) =γ(t_x)=x≠γ(t^*). (see again Claim <ref> (i)). If now k≥ 2, let us suppose that there exists i_0∈{1,2,…,k-1}, γ(t'_i_0)=γ(t^*). Then, from the definition of the term t_i_0+1, we obtain that t_i_0+1=t^* and t'_i_0+1=t^*, which results that, γ(t'_i_0+1)=γ(t^*). Again, from the definition of the term, t_i_0+2, we obtain that t_i_0+2=t^* and t'_i_0+2=t^*. Continuing this procedure for  finite steps until we reach the term t_k-1, and conclude that t_k-1=t^*=t'_k-1. Therefore, γ(t'_k-1)=γ(t^*) and as a consequence from the definition of t_k, we receive that t_k=t^*=t'_k=t^*. This leads to the fact that γ(t'_k)=γ(t^*) which contradicts our initial assumption. Therefore, γ(t^*)∈ M∖(∪_i=0^k {γ(t'_i)}). I.e t^*∈ J_k+1. From the last we conclude that, t_k+1=inf J_k+1 is well defined. Next, we have that, t'_k+1=sup J'_k+1 where J'_k+1={ t∈ [t'_k,t^*]: γ(t)=γ(t_k+1) }. Obviously, J'_k+1 is upper bounded from t^*. Moreover, J'_k+1≠∅, since t_k+1∈ J'_k+1. Indeed, since t'_k is a lower bound of J_k+1, it follows that t_k+1=inf J_k+1≥ t'_k. Now, due to the fact, that t^* is an upper bound of J_k+1, it follows that t_k+1≤ t^*. In conclution, t_k+1∈ [t'_k,t^*]. From the above analysis, through the Induction Principle, we receive the desired result. 
		 		
		 		
		 		
  (ii) Let k_0∈ randomly selected. We distinguish the following cases for γ(t'_k_0-1): Let γ(t'_k_0-1) =γ(t^*) then t_k_0=t^* and γ(t_k_0)=γ(t^*). From the Claim <ref> (i)  it follows γ(t^*) ∈ M, consequently γ(t_k_0)∈ M. Now let γ(t'_k_0-1)≠γ (t^*), then t_k_0=inf J_k_0.  From the fact that, t_k_0:=inf J_k_0 is a limit point of J_k_0, there exists a sequence (t̃_l)_l∈⊂ J_k_0 such that t̃_linf J_k_0≡ t_k_0. Consequently, from the continuity of γ we receive that, γ(t̃_l)γ(t_k_0). Since, (t̃_l)_l∈⊂ J_k_0, it follows that γ(t̃_l)∈ M, ∀ l∈. Therefore, γ(t_k_0)∈ M since M is closed. Next, we prove that, γ(t'_k_0)=γ(t_k_0)∈ M. Since t'_k_0=sup J'_k_0 and sup J'_k_0 is a limit point of J'_k_0 there exists a sequence (t̅_l )_l∈⊂ J'_k_0 such that t̅_lsup J'_k_0≡ t'_k_0. By the contuinity of γ, we receive γ(t̅_l)γ(t'_k_0). From the fact that (t̅_l )_l∈⊂ J'_k_0 we have that γ(t̅_l)=γ(t_k_0), ∀ l∈. By the uniqueness of limit, it follows that γ(t'_k_0) =γ(t_k_0)∈ M. 
		 		
		 		
  (iii) Let k_0∈ randomly selected. From the (ii), we conclude that t_x<t_k_0, since if t_x=t_k_0, then it follows that x=γ(t_x)=γ(t_k_0) i.e x∈ M which contradicts with the assumption that x∈ U∖ M. By the definition of t'_k_0 we obtain that, t'_k_0≤ t^*, since t^* is an upper bound of J'_k_0. Next, we prove that t_k_0≤ t'_k_0. Let γ(t'_k_0-1)=γ(t^*), then t_k_0=t^* and γ(t_k_0) =γ(t^*). Also from the last we obtain that t^*∈ J'_k_0. Consequently, since t^* is an upper bound of J'_k_0, we receive t'_k_0=sup J'_k_0=t^*  and t_k_0=t'_k_0=t^*. Let now γ(t'_k_0-1 )≠γ(t^*), then t_k_0=inf J_k_0. It holds that, t_k_0≥ t'_k_0-1 since t'_k_0-1 is a lower bound of J_k_0. Moreover, t^* is an upper bound of J_k_0, and that automatically gives that t_k_0≤ t^*. We conclude, that t'_k_0-1≤ t_k_0≤ t^* therefore t_k_0∈ J'_k_0. By the definition of t'_k_0≡sup J'_k_0, it follows that, t_k_0≤ t'_k_0.
		 		
		 		
  (iv) Let us suppose that, there exists k_0∈ such that t_k_0=t^*. The result will be proved through the Induction Principle. For k=k_0 we have, t_k_0=t^*. Then, t'_k_0=sup J'_k_0 where J'_k_0=sup{ t∈[t'_k_0-1,t^*]: γ(t)=γ(t^*)}. It holds that, t^*∈ J'_k_0 and t^* is an upper bound of J'_k_0. Therefore, t'_k_0=sup J'_k_0=max J'_k_0=t^*. Now, we suppose that for random k≥ k_0 that holds t_k=t^*=t'_k. We will prove that t_k+1=t^*=t'_k+1. Indeed since, t'_k=t^* it follows that γ(t'_k) =γ(t^*). Then from the definition of t_k+1, we obtain that t_k+1=t^*. Furthermore, t'_k+1=sup J'_k+1=sup{ t∈[t'_k,t^*]: γ(t)=γ(t^*) }= t^*, since t^* is an upper bound of J'_k+1 and also t^*∈ J'_k+1. In conclution, from the Induction Principle, we get the desired result.

		 		 Next, let k_0∈ such that, t'_k_0=t^*. Then by the definition of t_k_0+1 it follows that t_k_0+1=t^*. From the previous analysis, we receive that, ∀ k≥ k_0+1, t_k=t^*=t'_k.
		 		
		 		
  (v)
		 		
		 			
  (a)Let k^*∈. Initially, we observe that the desired inequality is true for k=1, since from Claim <ref>(ii) we get t_x<t^*.  Let us suppose the opposite of what we seek to prove i.e there exists k_0∈{2,…,k^*} such that t_k_0-1≥ t^*. We already know from Claim <ref>(iii) that t_k_0-1≤ t^*. Therefore, t_k_0-1=t^* and γ(t_k_0-1)=γ(t'_k_0-1)=γ(t^*). As a result from the last, we receive that k_0-1∈ J therefore k^*:=inf J≤ k_0-1, which is a contradiction, since k_0-1<k^*. With similar way we prove the desired inequality for the case where k^*=∞. Moreover, we prove with analogous way the inequality t'_k-1<t^*, ∀ k∈ K.
		 			
		 			
  (b)Let k^*∈.  Initially, we observe that the desired inequality is true for k=1, since, γ(t_0)=γ(t_x)=x∉ M by assumption. At the same time, we know from Claim <ref>(i) that γ(t^*)∈ M and as a consequence γ(t_0)≠γ(t^*). Let us suppose the opposite of what we seek to prove i.e there exists k_0∈{2,…,k^*} such that γ(t_k_0-1)=γ( t^*). Clearly the last contradicts with the definition of k^*, since k_0-1<k^*.  With similar way we prove the desired inequality for the case where k^*=∞.
		 			
		 			
  (c)  Let k^*∈ and we select randomly k_0∈{1,2,…,k^*}. We will prove that, t'_k_0-1<t_k_0. For this, we distinguish the following cases for t_k_0.
		 			
		 				
  * ( t_k_0=t^*) Then from Claim <ref> v(a) we obtain that t'_k_0-1<t^*=t_k_0 since k_0-1<k_0≤ k^* therefore, t'_k_0-1< t_k_0.
		 				
  * (t_k_0<t^*) From the Claim <ref> v(b) we obtain that γ(t'k_0-1) ≠γ(t^*). In consequence, t_k_0=inf J_k_0 where J_k_0:= { t∈[t'_k_0-1,t^*]: γ(t)∈ M∖(∪_i=0^k_0-1{γ(t'_i)})  }. It is clear that t'_k_0-1 is a lower bound of J_k_0 and as a result by the definition of t_k_0=inf J_k_0, we receive that t'_k_0-1≤ t_k_0. In specific, we will prove that t'_k_0-1<t_k_0. Let t'_k_0-1=t_k_0 therefore, x_k_0-1=γ(t_k_0-1)=γ(t'_k_0-1)=γ(t_k_0). From the assumption of Theorem  <ref>, it holds that x_k_0-1∈ M=iso(M). In consequence, there exists δ_k_0-1>0 such that B_d(x_k_0-1,δ_k_0-1)∩ M∖{x_k_0-1}=∅. Due to the fact that, t_k_0=inf J_k_0 is a limit point of the set J_k_0, there exists a sequnece ( t̃_l)_l∈⊂ J_k_0 such that t̃_lt_k_0. Furthermore, from the continuity of γ, it holds that γ(t̃_l)γ(t_k_0). Now, from the definition of J_k_0, since ( t̃_l)⊂ J_k_0 it follows that ∀ l∈, γ(t̃_l)∈ M∖(∪_i=0^k_0-1{γ(t'_i)}) ⊂ M∖{γ(t'_k_0-1)}=M∖{x_k_0-1}. From the convergence of (γ(t̃_l) )_l∈, there exists l_0∈, such that, ∀ l≥ l_0, γ(t̃_l)∈ B_d(γ(t_k_0) ,δ_k_0-1)=B_d( x_k_0-1,δ_k_0-1). From the last, we conclude that, B_d( x_k_0-1,δ_k_0-1)∩ M∖{x_k_0-1}≠∅ which contradicts with the fact that B_d(x_k_0-1,δ_k_0-1)∩ M∖{x_k_0-1}=∅. In a similar way we prove the desired inequality for the case where k^*=∞.
		 			
		 			
		 			
  (d) Let k^*∈ and we select randomly k_0∈{1,2,…,k^*}. Then from Claim <ref> (iii), we obtain that t_k_0-1≤ t'_k_0-1. But from Claim <ref> v(c) we have also  t'_k_0-1<t_k_0. Combining the last two, we receive the desired inequality. In a similar way we prove the desired inequality for the case where k^*=∞.
		 			
		 			
  (e) Let k^*∈ and we select randomly k_0∈{1,2,…,k^*}. Then from Claim <ref> v (c), we obtain that t'_k_0-1< t_k_0. But from Claim <ref> (iii) we have also  t_k_0≤ t'_k_0. Combining the last two, we receive the desired inequality. In a similar way we prove the desired inequality for the case where k^*=∞.
		 		
	 		  
	 		    
  (vi) Let k^*∈. We suppose the opposite of what we seek to prove i.e there are i_0,j_0∈ K:={1,2,…,k^*} such that x_i_0=γ(t_i_0)=γ(t_j_0):=x_j_0. Without loss of generality, let i_0<j_0. If j_0=k^* then γ(t_j_0)=γ(t_k^*)=γ(t^*). But from the Claim <ref>v(b) we conclude that γ(t_i_0)≠γ(t^*)=γ(t_j_0), since i_0<j_0=k^*. The last result contracicts to our initial assumption γ(t_i_0)=γ(t_j_0). Now let j_0<k^*. By the assumption of theorem, we know that x_i_0∈ iso(M). From the last, we obtain that there exists δ_i_0>0 such that B_d(x_i_0,δ_i_0)∩ M∖{ x_i_0}=∅. Since j_0-1<k^*, from the Claim <ref>v(b) we receive γ(t_j_0-1)≠γ(t^*). Therefore, t_j_0=inf J_j_0 where J_j_0:=inf{ t∈[t'_j_0-1,t^*]: γ(t)∈ M∖( ∪_i=0^j_0-1{γ(t'_i)}) }. Since inf J_j_0 is a limit point of J_j_0, there exists a sequence  (t̃_l )_l∈⊂ J_j_0 such that t̃_linf J_j_0≡ t_j_0. Moreover, from the continuity of γ, we obtain that γ(t̃_l)γ(t_j_0). Now, by the definition of J_j_0, it follows that ∀ l∈, γ(t̃_l)∈ M∖(∪_i=0^j_0-1{γ(t'_i)})⊂ M∖{γ(t'_i_0)} where the last inclusion holds from the fact i_0<j_0 i_0≤ j_0-1. From the convergence of (γ( t̃_l)  )_l∈ there exists l_0∈ such that for all l≥ l_0, γ(t̃_l)∈ B_d(γ(t_j_0),δ_i_0) =B_d(γ(t_i_0),δ_i_0). Consequently, ∅≠ B_d(γ(t_i_0),δ_i_0)∩ M∖{γ(t'_i_0)}=B_d(γ(t_i_0),δ_i_0)∩ M∖{γ(t_i_0)} where the last contradicts with fact that B_d(x_i_0,δ_i_0)∩ M∖{ x_i_0}=∅.  In a similar way we prove the desired inequality for the case where k^*=∞.
	 		    
	 		    
  (vii) Let k^*∈. Let us suppose the opposite of what we seek to prove i.e. there exists k_0∈ K:={1,2,…,k^*} such that γ( ( t'_k_0-1,t_k_0) ) ∩ M≠∅. Therefore, there exists z∈ M such that z=γ(s) with t'_k_0-1<s<t_k_0. From the Claim <ref> v (b) it holds that γ(t'_k_0-1)≠γ(t^*) since k_0-1<k^*. As a result, t_k_0=inf J_k_0. We claim that s∈ J_k_0. For the last it is sufficent to prove that, γ(s)≠γ(t'_i), ∀ i∈{0,1,…,k_0-1}. If i=0 then z:=γ(s)≠γ(t'_0)=γ(t_x):=x since x∈ U∖ M and  z∈ M. It remains to prove γ(s)≠γ(t'_i), ∀ i∈{1,…,k_0-1} with k_0≥ 2. Indeed, let us suppose that the last does not hold then there exists l∈{1,2,…,k_0-1} such that γ(s)=γ(t'_l). We have that t'_l:=sup J'_l, where J'_l:={ t∈[t'_l-1,t^*]: γ(t)=γ(t_l) }. Since, t'_l-1≤^(Claim <ref> v(e) )  t'_k_0-1<s<t_k_0<^( Claim<ref> (iii))t^* we obtain that s∈ J'_l. Therefore, t'_l≥ s. But the last contradicts with the fact that t'_l≤^(Claim <ref> v(e) )  t'_k_0-1<s. Consequently, we receive that z=γ(s)∈ M∖(∪_i=0^k_0-1{γ(t'_i) }) and since s∈( t'_k_0-1,t_k_0), it follows that s∈ J_k_0. Therefore, t_k_0=inf J_k_0≤ s. The last leads to a contradiction since s<t_k_0. In a  similar way we prove the desired inequality for the case where k^*=∞.
 		  		 	
 	  		 			
		 
			 
    We define, x_k:=γ(t_k), k∈ K and x^*:=γ(t^*). Let k∈ K. From the assumption, x_k∈ int(U) and x_k∈ iso(M). Therefore, there exist ϵ_x_k>0,δ̃_x_k>0 such that B_d( x_k,ϵ_x_k)⊂ U and B_d( x_k,δ̃_x_k) ∩ M={x_k}.	
	For the construction of the path γ̃:[t_x,t_y]→ U∖ M, connecting the points x, y, we distinguish the following cases for k^*.

	Case: k^*∈

		 Equivalently, we have  K:={1,2,…,k^*}. In this case, we have x_k^*:=γ(t_k^*)=γ(t^*)=x^*.

		 We begin the construction of parts of the path γ̂:[t_x,t_y]→ U∖ M.
		 
	
		
  (i) (k^*=1). Then K={1} and γ(t_1)=γ(t'_1)=γ(t^*)≡ x^*. 

		We define:
		
    t̃^*:=inf{ t∈[t_x,t_1]: γ(t)=γ(t_1)}≡infS̃_k^*.

		We notice that, S̃_k^*≠∅, since t_1∈S̃_k^*. Moreover, S̃_k^* is lower bounded by t_x thus t̃^* is well defined. Furthermore, by the continuity of γ, we obtain that γ(t̃^*)=γ(t_1). We claim also that t_1=t̃^*. Indeed, from the Claim <ref> (iii) we obtain t_x<t_1. In addition, from Claim <ref> (vii), we obtain that γ( (t'_0,t_1 ) )∩ M=γ( (t_x,t_1 ) )∩ M=∅. In consequence, ∀ z∈ (t_x,t_1), γ(z)∉ M and as a result γ(z)≠ x^*∈ M. From the last, we receive that, S̃_k^*={t_1} resulting that, t_1=t̃^*.      
	Next, we define:
		
    δ_x_k^*:=1/2min{ϵ_x_k^*,δ̃_x_k^*,d(x_k^*,x),d(x_k^*,y)}.
 
	 Clearly, we see that,
	 	
    δ_x_k^*≤δ'_x_k^*:=1/2min{ϵ_x_k^*,d(x_k^*,x) } andδ_x_k^*≤δ”_x_k^*:=1/2min{ϵ_x_k^*,d(x_k^*,y) }

	where x_k^*≠ x and x_k^*≠ y, since x_k^*∈ M and M∩{x,y}=∅. Furthermore, t_x<^( Claim <ref>(iii)) t_1=t̃^* and t^*<t_y
	Then, from the Proposition <ref>, ∀δ∈(0,δ_x_k^*]
	
    ∅≠ K'_x_k^*:=γ([t_x,t̃^*])  ∩∂ B_d( x_k^*,δ) ⊂ U∖ M and∅≠ K”_x_k^*:=γ([ t^*,t_y])  ∩∂ B_d( x_k^*,δ) ⊂ U∖ M.

      We fix such δ_k^*∈ (0,δ_x_k^*). Therefore, 
      
    ∃ z'_k^*∈[t_x,t̃^*]: γ(z'_k^*)∈∂ B_d(x_k^*,δ_k^*), ∃ z”_k^*∈[t^*,t_y]: γ(z”_k^*)∈∂ B_d(x_k^*,δ_k^*).

      We define,
      
    t_x'_k^*:=inf{ t∈[t_x,t̃^*]: γ(t)∈∂ B_d(x_k^*,δ_k^*) }≡inf S_k^*,
    
          	t_x”_k^*:=sup{ t∈[t^*,t_y]: γ(t)∈∂  B_d(x_k^*,δ_k^*)}≡sup S'_k^*
    
          	x'_k^*:=γ(t_x'_k^*), and x”_k^*:=γ(t_x”_k^*).

       Next, we observe that S_k^*≠∅, since z'_k^*∈ S_k^* and S'_k^*≠∅, since z”_k^*∈ S'_k^*. Moreover, S_k^* is lower bounded from t_x and S'_k^* is upper bounded by t_y. As a result from the last, t_x'_k^*, t_x”_k^* are well defined. We mention that, by their definition of t_x'_k^*, t_x”_k^*, we receive also, t_x≤ t_x'_k^* and t_x”_k^*≤ t_y.  Furthermore, using the continuity of γ and the fact that ∂ B_d(x_k^*,δ_k^*) is closed set, we conclude that x'_k^*≡γ(t_x'_k^*)∈∂ B_d(x_k^*,δ_k^* )⊂ U∖ M and x”_k^*≡γ(t_x”_k^*)∈∂ B_d(x_k^*,δ_k^*)⊂ U∖ M.[We know that δ_k^*<ϵ_x_k^* and δ_k^*<δ̃_x_k^*. Therefore, ∂ B_d(x_k^*,δ_x_k^*)⊂B̅_d(x_k^*,δ_k^*)⊂ B_d(x_k^*,ϵ_x_k^*)⊂ U and ∂ B_d(x_k^*,δ_k^* )⊂B̅_d(x_k^*,δ_k^*)⊂ B_d(x_k^*,δ̃_x_k^*). Therefore, ∂ B_d(x_k^*,δ_k^* )∩ M⊂ B_d(x_k^*,δ̃_x_k^*)∩ M={x_k^*} but ∂ B_d(x_k^*,δ_k^* )∩{x_k^*}=∅. In consequence, ∂ B_d(x_k^*,δ_k^* )∩ M=∅. We conclude then, ∂ B_d(x_k^*,δ_k^* )⊂ U∖ M.] From the last, it is extracted that t_x<t_x'_k^*, t_x”_k^*<t_y. Indeed, since x=γ(t_x)∈ B_d(x,1/2d(x,x^*)) and x'_k^*=γ(t_x'_k^*)∈ B_d(x_k^*,δ_k^*)⊂ B_d(x_k^*,1/2d(x,x_k^*)) where, B_d(x,1/2d(x,x_k^*))∩ B_d(x^*,1/2d(x,x_k^*))=∅. Therefore, t_x≠ t_x'_k^* and as  result from above t_x≤ t_x'_k^*, we receive that t_x< t_x'_k^*. Similarly, we prove that t_x”_k^*<t_y.  Using now the assumption of theorem, the boundary of ball B_d(x_k^*,δ_k^*)⊂ U∖ M is path-connected. Therefore, there exists a continuous function γ̃_x'_k^*,x”_k^*:[t_x'_k^*,t_x”_k^*]→∂ B_d(x_k^*,δ_k^*) such that, γ̃_x'_k^*,x”_k^*(t_x'_k^*) =x'_k^* and γ̃_x'_k^*,x”_k^*(t_x”_k^*) =x”_k^*. Notice that, from Claim <ref>, γ( t^*,t_y)∩ M=∅, resulting that γ( (t_x”_k^*,t_y ))  ∩ M=∅. Another important information, is the fact that, γ( (t_x,t_x'_k^*))  ∩ M=∅. Indeed, this is obvious from the fact that, (t_x,t_x'_k^*)⊂(t_x,t̃^*)=(t_x,t_1) and γ(( t_x,t_1)  )∩ M=∅, from the Claim <ref> (vii).  

      We define the path γ̃:[t_x,t_y]→ U∖ M, as follows:
      
    γ̃(t):=γ(t),     t∈[t_x,t_x'_k^*]
    γ̃_x'_k^*,x”_k^*(t),     t∈[t_x'_k^*,t_x”_k^*]
    γ(t),     t∈[t_x”_k^*,t_y].

      

       
  (ii) (k^*≥ 2), equivalently, {1,2}⊂ K.

		We define, ∀ k∈ K,
		
    r_k:=min{ d(x_k,x_i): i∈ K∖{k}} andδ_x_k:=1/2min{ϵ_x_k,δ̃_x_k,d(x_k,x),d(x_k,y),r_k }.

		We make the following observations for the open balls B_d( x_k,δ_x_k), k∈ K.
		
			
  (i) ∀ k∈ K, ∀δ∈(0,δ_x_k], B̅_d( x_k,δ)⊂ U 
			
  (ii) ∀ k∈ K, ∀δ∈(0,δ_x_k], B̅_d( x_k,δ)∩ M={x_k}
			
  (iii)  ∀ k∈ K,  ∀δ∈(0,δ_x_k], B̅_d( x_k,δ)∩{ x,y}  =∅
			
  (iv)  ∀ i,j∈ K, i≠ j, ∀δ∈(0,δ_x_k],∀δ̃∈(0,δ_x_j], B_d( x_i,δ)∩ B_d(x_j,δ̃)=∅
			
  (vi) ∀ k∈ K,  ∀δ∈(0,δ_x_k], ∂ B_d( x_k,δ)⊂ U∖ M.
		

		 Clearly we see that, 
		
    δ_x_1≤δ'_x_1:=1/2min{ϵ_x_1,d(x_1,x) } andδ_x_1≤δ”_x_1:=1/2min{ϵ_x_1,d(x_1,x_2) }.

		Then Proposition <ref>, ∀δ∈(0,δ_x_1]
		
    ∅≠ K'_x_1:=γ([t_x,t_1])  ∩∂ B_d( x_1,δ) ⊂ U∖ M and∅≠ K”_x_1:=γ([ t'_1,t_2])  ∩∂ B_d( x_1,δ) ⊂ U∖ M.

		We fix such δ_1∈(0,δ_x_1). Therefore, 
		
    ∃ z'_1∈[t_x,t_1]: x'_1:=γ(z'_1)∈∂ B_d(x_1,δ_1 ), ∃ z”_1∈[t'_1,t_2]: x”_1:=γ(z”_1)∈∂ B_d(x_1,δ_1 ).

		We define 
		
    t_x'_1:=inf{ t∈[t_x,t_1]: γ(t)∈∂ B_d(x_1,δ_1) }≡inf S_1,
    
    			t_x”_1:=sup{ t∈[t'_1,t_2]: γ(t)∈∂  B_d(x_1,δ_1)}≡sup S'_1
    
    			x'_1:=γ(t_x'_1), and x”_1:=γ(t_x”_1).

		First, we observe that S_1≠∅, since z'_1∈ S_1 and S'_1≠∅, since z”_1∈ S'_1. Moreover, S_1 is lower bounded by t_x and S'_1 is upper bounded by t_2. As a result from the last, t_x'_1, t_x”_1 are well defined. Furthermore, using the continuity of γ and the fact that ∂ B_d(x_1,δ_1) is closed set, we conclude that, γ(t_x'_1)∈∂ B_d(x_1,δ_1 )⊂ U∖ M, and γ(t_x”_1)∈∂ B_d(x_1,δ_1)⊂ U∖ M.  Now, we claim that t_x'_1<t_1≤ t'_1<t_x”_1. Obviously, it holds t_1≤ t'_1 from Claimi <ref>(iii). At first, we notice that t_1 is an upper bound of S_1 and t'_1 is a lower bound of S̃_1. Therefore, from the definition of t_x'_1 and t_x”_1, it follows that t_x'_1≤ t_1 and t'_1≤ t_x”_1. Next we observe that,
		
    x'_1≡γ(t_x'_1)∈∂ B_d(x_1,δ_1) and x”_1≡γ(t_x”_1)∂ B_d(x_1,δ_1)
     
    		x_1≡γ(t_1)=γ(t'_1)∉∂ B_d(x_1,δ_1).

		From the last, we receive that, t_x'_1<t_1 and t'_1<t_x”_1.
		 Using now the assumption of theorem, the boundary of ball B_d(x_1,δ_1 )⊂ U∖ M is path-connected. Therefore, there exists a continuous function γ̃_x'_1,x”_1:[t_x'_1,t_x”_1]→∂ B_d(x_1,δ_1 )⊂ U∖ M such that, γ̃_x'_1,x”_1(t_x'_1) =x'_1 and γ̃_x'_1,x”_1(t_x”_1) =x”_1. Another information to mention, is the fact that, γ((t_x,t_x'_1) )∩ M=∅. Indeed, since (t_x,t_x'_1)⊂(t_x,t_1) and from Claim <ref> v(ii), we know that γ((t_x,t_1) )∩ M=∅, we conclude that, γ((t_x,t_x'_1) )∩ M=∅.
		

		The next step is the extraction of appropriate intermediate points (t_x'_k)_k=2,…,k^*, (t_x”_k)_k=2,…,k^*⊂ [t_x,t_y], using the Principle of Induction. In specific, we define inductively the following sequences of points (t_x'_k)_k=2,…,k^* and  (t_x”_k)_k=2,…,k^* such that the corresponding sequences ( γ(t_x'_k))_k=2,…,k^*, (γ(t_x”_k) )_k=2,…,k^*⊂∂ B_d(x_k,δ_k) where δ_k appropriate radious. We will prove the existence of the above sequences inductively.
		
			Let, k^*∈, with k^*≥ 2. Then, for every k∈ K∖{1}={2,…,k^*}, there exists (t_x'_i)_i=2,…,k, (t_x”_i)_i=2,…,k⊂ [t_x,t_y] and δ_i∈  (0,δ_x_i), ∀ i∈{2,…,k} such that:
			
				
  (i) If k∈ K with k^*≥ 3 then, ∀ i∈{2,3,…,k-1}, t_x”_i-1<t_x'_i<t_i≤ t'_i<t_x”_i<t_i+1 and if k=k^* then t_x”_k^*<t_y. 
				
  (ii)  ∀ i∈{2,3,…,k}, x'_i:=γ(t_x'_i), x”_i:=γ(t_x”_i)∈∂ B_d(x_i,δ_i)⊂ U∖ M
				
  (iii) ∀ i∈{2,3,…,k}, γ( (t_x”_i-1,t_x'_i) )∩ M=∅, equivalently, γ( (t_x”_i-1,t_x'_i) )⊂ U∖ M
				
  (iv)If k=k^*, then γ( (t_x”_k^*,t_y)) ∩ M=∅.
				
  (v) For all i∈{1,2,…, k}, there exists a continuous function γ̃_x'_k,x”_k:[t_x'_k,t_x”_k]→∂ B_d(x_k,δ_k ) such that, γ̃_x'_k,x”_k(t_x'_k) =x'_k and γ̃_x'_k,x”_k(t_x”_k) =x”_k.
			
		
		
		
			We prove this by Induction. In specific, for k=2, we set δ^*_x_2:=1/2min{δ_x_2,d(x_2,x”_1)}<δ_x_2. We claim that δ^*_x_2>0, equivalently, d(x_2,x”_1)>0, since we already know that, δ_x_2>0.  Indeed, x_2∈ B_d( x_2,δ_x_2) and x”_1∈∂ B_d( x_1,δ_1)⊂ B_d( x_1,δ_x_1). From property, (iv) for open balls, we receive that, B_d( x_2,δ_x_2) ∩  B_d( x_1,δ_x_1)=∅. Therefore, x_2≠ x”_1. We observe that,
			
    δ^*_x_2≤1/2min{ϵ_x_2,d(x_2,x”_1) } and δ^*_x_2≤1/2min{ϵ_x_2,d(x_2,x_3) },     k^*≥ 3 
    1/2min{ϵ_x_2,d(x_2,y) },     k^*=2 .

			Now, from Proposition <ref>, ∀δ∈(0,δ^*_x_2]
			
    ∅≠ K'_x_2:=γ([t_x”_1,t_2]) ∩∂ B_d( x_2,δ) ⊂ U∖ M and
    ∅≠ K”_x_2:=γ([t'_2,t_3]) ∩∂ B_d( x_2,δ) ⊂ U∖ M,     k^*≥ 3
    γ([t^*,t_y]) ∩∂ B_d( x_2,δ) ⊂ U∖ M,     k^*=2 .

			We fix such δ_2∈(0,δ^*_x_2). As a result from the last,
			
    ∃ z'_2∈[t_x”_1,t_2], γ(z'_2)∈∂ B_d(x_2,δ_2 ), ∃ z”_2∈
    					[t'_2,t_3],     k^*≥ 3
    
    					[t^*,t_y],     k^*=2, γ(z”_2)∈∂ B_d(x_2,δ_2 ).
 
			Next, we define
			
    t_x'_2   :=inf{ t∈[t_x”_1,t_2]: γ(t)∈∂ B_d(x_2,δ_2)}≡inf S_2
        
        				t_x”_2   :=sup{ t∈[t'_2,t_3]: γ(t)∈∂  B_d(x_2,δ_2)}≡sup S'^(k^*≥ 3)_2,     k^*≥ 3
        sup{ t∈[t^*,t_y]: γ(t)∈∂  B_d(x_2,δ_2)}≡sup S'^(k^*=2)_2,     k^*=2.

			

			We claim that t_x'_2,t_x”_2 are well defined. Indeed, S_2≠∅, since z'_2∈ S_2 and S'_2≠∅, since z”_2∈ S'^(k^*≥ 3)_2,
(  z”_2∈ S'^(k^*=2)_2). Moreover, S_2 is lower bounded by t_x”_1 and S'^(k^*≥ 3)_2 is upper bounded by t_3. Also S'^(k^*=2)_2 is upper bounded by t_y. Therefore, t_x'_2,t_x”_2 are well defined.  Furtheromore, using the continuity of γ and the fact that ∂ B_d(x_2,δ_2) is closed set, we conclude that, x'_2≡γ(t_x'_2)∈∂ B_d(x_2,δ_2), and x”_2≡γ(t_x”_2)∈∂ B_d(x_2,δ_2).  As we did in the proof for the case where k=1, we obtain again that t_x'_2<t_2≤ t'_2<t_x”_2.
			Now, we proceed to the proof of t_x”_1<t_x'_2. First, we notice that t_x”_1 is a lower bound of S_2 and as a result from the definition of infimum, we obtain that, t_x”_1≤ t_x'_2. We claim, that t_x”_1< t_x'_2. For this, it is sufficent to prove that t_x”_1≠ t_x'_2, since t_x”_1 is a lower bound of S_2. At first, we see that, the following holds:
			
    x”_1∈∂ B_d(x_1,δ_1)⊂B̅_d(x_1,δ_1)⊂ B_d(x_1,δ_x_1), since δ_1≤δ_x_1
    
    				x'_2∈∂B̅_d(x_2,δ_2)⊂ B_d(x_2,δ_x_2), since δ_2≤δ^*_x_2<δ_x_2.

			Using now property, (iv), we obtain that, B_d(x_1,δ_x_1)∩ B_d(x_2,δ_x_2)=∅. Therefore, x”_1≠ x'_2, resulting that, t_x”_1≠ t_x'_2, which leads to t_x”_1< t_x'_2.	Moreover, we claim that t_x”_2<t_3 in case where k^*≥ 3. Since, t_3 is an upper bound of S'^(k^*≥ 3)_2, it follows that t_x”_2≤ t_3. It remains to prove that, t_x”_2≠ t_3. Indeed, we observe that:
			
    x”_2∈∂ B_d(x_2,δ_2) ⊂B̅_d(x_2,δ_2)⊂ B_d(x_2,δ_x_2) and x_3=γ(t_3)∈ B_d(x_3,δ_x_3).

			Using now property (iv), we obtain that B_d(x_2,δ_x_2)∩ B_d(x_3δ_x_3). Therefore, x”_2≠ x_3, resulting that t_x”_2≠ t_3. If now, k^*=2,  we prove similarly that t_x”_2<t_y. In addition, γ( (t_x”_1,t_x'_2))∩ M=∅. Indeed, this is true from the fact that γ((t'_1,t_2) )∩ M=∅, (see Claim <ref> (vii)) and (t_x”_1,t_x'_2)⊂(t_x”_1,t_2). Also in case, where k^*=2, then γ( (t_x”_2,t_y))∩ M=∅, since (t_x”_2,t_y)⊂ (t^*,t_y) and from Claim <ref> (iii), γ((t^*,t_y))∩ M=∅. From the assumption of theorem, we know that the boundary of the ball B_d(x_2,δ_2)⊂ U∖ M is path-connected. Therefore, there exists a continuous function γ̃_x'_2,x”_2:[t_x'_2,t_x”_2]→∂ B_d(x_2,δ_2) such that γ̃_x'_2,x”_2(t_x'_2) =x'_2 and γ̃_x'_2,x”_2(t_x”_2) =x”_2.
			

			Next, let us suppose that for a random k'∈{2,…,k^*-1} with k^*≥ 3, we generated the following sequences, (t_x'_i)_i=2,…,k',  (t_x”_i)_i=2,…,k'⊂ [t_x,t_y] and also δ∈(0,δ_x_i), ∀ i_i=2,…,k' which satisfy properties (i)-(v) of the Claim. (Inductive Step). We set, x'_i:=γ(t_x'_i),x”_i:=γ(t_x”_i), ∀ i∈{2,3,…,k'}. The target now, is to prove the statement in the case where k=k'+1. In order to prove the last, it is sufficcent from the Inductive step, to prove the existence of terms t_x'_k'+1, t_x”_k'+1⊂ I and δ_k'+1⊂ (0,δ_x_k'+1) satisfying the following:
			
			
  * If,	k'+1<k^*,then t_x”_k'<t_x'_k'+1<t_k'+1≤ t'_k'+1<t_x”_k'+1 and if k'+1=k^*,then t_x”_k'+1<t_y 
			
  * γ(t_x'_k'+1), γ(t_x”_k'+1)∈∂ B_d(x_k'+1,δ_k'+1)
			
  * γ((t_x”_k',t_x'_k'+1)) ∩ M=∅
			
  * If k'+1=k^*, then γ( (t_x”_k'+1,t_y))∩ M=∅
			
  * There exists a continuous path γ̃_x'_k'+1,x”_k'+1:[t_x'_k'+1,t_x”_k'+1]→∂ B_d(x_k'+1,δ_k'+1), with 
 γ̃_x'_k'+1,x”_k'+1(t_x'_k'+1) =x'_k'+1 and γ̃_x'_k'+1,x”_k'+1(t_x”_k'+1)=x”_k'+1.
			
			We set δ^*_x_k'+1:=1/2min{δ_x_k'+1,d(x_k'+1,x”_k') }. We claim that δ^*_x_k'+1>0, equivalently, d(x_k'+1,x”_k')>0, since we already know that δ_x_k'+1>0. lndeed, x_k'+1∈ B_d(x_k'+1,δ_x_k'+1) and x”_k'∈∂ B_d(x_k',δ_k')⊂ B_d(x_k',δ_x_k'). From property, (iv) for open balls, we receive that B_d(x_k'+1,δ_x_k'+1)∩ B_d(x_k',δ_x_k')=∅. Therefore, x_k'+1≠ x”_k'. Moreover, 
			
    δ^*_x_k'+1≤1/2min{ϵ_x_k',d(x_k'+1,x”_k')} and δ^*_x_k'+1≤1/2min{ϵ_x_k',d(x_k'+1,x_k'+2)},     k'+1<k^*
    1/2min{ϵ_x_k',d(x_k'+1,y)},     k'+1=k^*.

			Now, from Proposition <ref>, ∀δ∈(0,δ^*_x_k'+1]
			
    ∅≠ K'_x_k'+1:=γ( [t_x”_k',t_k'+1] )∩∂ B_d(x_k'+1,δ)⊂ U∖ M and
    ∅≠ K”_x_k'+1:=γ( [t'_k'+1,t_k'+2] )∩∂ B_d(x_k'+1,δ)⊂ U∖ M,     k'+1<k^*
    γ( [t^*,t_y] )∩∂ B_d(x_k'+1,δ)⊂ U∖ M,     k'+1=k^*.

			We fix such δ_k'+1∈(0,δ^*_x_k'+1 ). As a result from the last, 
			
    ∃ z'_k'+1∈[t_x”_k',t_k'+1], γ(z'_k'+1)∈∂ B_d(x_k'+1,δ_k'+1),
    ∃ z”_k'+1∈
    					[t'_k'+1,t_k'+2],     k'+1<k^*
    
    					[t^*,t_y],    k'+1=k^*
    				, γ(z”_k'+1)∈∂ B_d(x_k'+1,δ_k'+1).

			Next, we define
			
    t_x'_k'+1   :=inf{ t∈[t_x”_k',t_k'+1]: γ(t)∈∂ B_d(x_k'+1,δ_k'+1) }≡inf S_k'+1
    
    				t_x”_k'+1   :=sup{ t∈[t'_k'+1,t_k'+2]: γ(t)∈∂ B_d(x_k'+1,δ_k'+1) }≡sup S'^(k'+1<k^*)_k'+1,     k'+1<k^*
    sup{ t∈[t^*,t_y]: γ(t)∈∂ B_d(x_k'+1,δ_k'+1) }≡sup S'^(k'+1=k^*)_k'+1,     k'+1=k^*.

		
		We set, x'_k'+1:=γ(t_x'_k'+1) and x”_k'+1:=γ(t_x”_k'+1).
		We claim that t_x'_k'+1,t_x”_k'+1 are well defined. Indeed, S_k'+1≠∅, since z'_k'+1∈ S_k'+1 and S'^(k'+1<k^*)_k'+1≠∅, since z”_k'+1∈ S'^(k'+1<k^*)_k'+1,
(  z”_k'+1∈ S'^(k'+1=k^*)_2). Moreover, S_k'+1 is lower bounded by t_x”_k' and S'^(k'+1<k^*)_k'+1 is upper bounded by t_k'+2. Also S'^(k'+1=k^*)_k'+1 is upper bounded by t_y. Therefore, t_x'_k'+1,t_x”_k'+1 are well defined.  Furthermore, using the continuity of γ and the fact that ∂ B_d(x_k'+1,δ_k'+1) is closed set, we conclude that, x'_k'+1≡γ(t_x'_k'+1)∈∂ B_d(x_k'+1,δ_k'+1), and x”_k'+1≡γ(t_x”_k'+1)∈∂ B_d(x_k'+1,δ_k'+1).  As we did in the proof for the case where k=1, we obtain again that t_x'_k'+1<t_k'+1≤ t'_k'+1<t_x”_k'+1.
		Now, we proceed to the proof of t_x”_k'<t_x'_k'+1. First, we notice that t_x”_k' is a lower bound of S_k'+1 and as a result from the definition of infimum, we obtain that, t_x”_k'≤ t_x'_k'+1. We claim, that t_x”_k'< t_x'_k'+1. For this, it is sufficent to prove that t_x”_k'≠ t_x'_k'+1, since t_x”_k' is a lower bound of S_k'+1. At first, we see that, the following holds:
		
    x”_k'∈∂ B_d(x_k',δ_k')⊂B̅_d(x_k',δ_k')⊂ B_d(x_k',δ_x_k'), since δ_k'<δ_x_k'
    
    			x'_k'+1∈∂B̅_d(x_k'+1,δ_k'+1)⊂ B_d(x_k'+1,δ_x_k'+1), since δ_k'+1≤δ^*_x_k'+1<δ_x_k'+1.

		Using now property, (iv), we obtain that, B_d(x_k',δ_x_k')∩ B_d(x_k'+1,δ_x_k'+1)=∅. Therefore, x”_k'≠ x'_k'+1, resulting that, t_x”_k'≠ t_x'_k'+1, which leads to t_x”_k'< t_x'_k'+1.	Moreover, we claim that t_x”_k'+1<t_k'+2 in case where k^*≥ 3. Since, t_k'+2 is an upper bound of S'^(k'+1<k^*)_k'+1, it follows that t_x”_k'+1≤ t_k'+2. It remains to prove that, t_x”_k'+1≠ t_k'+2. Indeed, we observe that:
		
    x”_k'+1∈∂ B_d(x_k'+1,δ_k'+1) ⊂B̅_d(x_k'+1,δ_k'+1)⊂ B_d(x_k'+1,δ_x_k'+1) and x_k'+2=γ(t_k'+2)∈ B_d(x_k'+2,δ_x_k'+2).

		Using now property (iv), we obtain that B_d(x_k'+1,δ_x_k'+1)∩ B_d(x_k'+2δ_x_k'+2). Therefore, x”_k'+1≠ x_k'+2, resulting that t_x”_k'+1≠ t_k'+2. If now, k^*=k'+1,  we prove similarly that t_x”_k'+1<t_y. In addition, γ( (t_x”_k',t_x'_k'+1))∩ M=∅. Indeed, this is true from the fact that γ((t'_k',t_k'+1) )∩ M=∅, (see Claim <ref> (vii)) and (t_x”_k',t_x'_k'+1)⊂(t_x”_k',t_k'+1). Also in case, where k^*=k'+1 then γ( (t_x”_k'+1,t_y))∩ M=∅, since (t_x”_k'+1,t_y)⊂ (t^*,t_y) and from Claim <ref> (iii), γ((t^*,t_y))∩ M=∅. From the assumption of theorem, we know that the boundary of the ball B_d(x_k'+1,δ_k'+1)⊂ U∖ M is path-connected. Therefore, there exists a continuous function γ̃_x'_k'+1,x”_k'+1:[t_x'_k'+1,t_x”_k'+1]→∂ B_d(x_k'+1,δ_k'+1) such that, γ̃_x'_k'+1,x”_k'+1(t_x'_k'+1) =x'_k'+1 and γ̃_x'_k'+1,x”_k'+1(t_x”_k'+1) =x”_k'+1. By the use of Induction Principle, we conclude the existence of a finite sequence of points (t_x'_i)_i=2,…, k^*,  (t_x”_i)_i=2,…, k^*∈ I and δ_i∈(0,δ_x_i), ∀ i=2,…,k^* which satisfy properties (i)-(v) of the Claim.

    
	Finally, we define for the case where k^*≥ 2, the path γ̂:[t_x,t_y]→ U∖ M  as follows:
	
    γ̂(t):=γ(t),     t∈[t_x,t_x'_1]
    γ̃_x'_1,x”_1(t),     t∈[t_x'_1,t_x”_1]
    γ(t),     t∈[t_x”_1,t_x'_2]
    ⋮    ⋮
    γ(t),     t∈[t_x”_k-1,t_x'_k]
    γ̃_x'_k,x”_k(t),     t∈[t_x'_k,t_x”_k]
    γ(t),     t∈[t_x”_k,t_x'_k+1]
    ⋮    ⋮
    γ(t),     t∈[t_x”_k^*-1,t_x'_k^*]
    γ̃_x'_k^*,x”_k^*(t),     t∈[t_x'_k^*,t_x”_k^*]
    γ(t),     t∈[t_x”_k^*,t_y].





	Case: k^*=∞

	Equivalently we have  K:=. A direct concequence,for that case, is x_k≠ x^*, ∀ k∈. In this setting we define again with a similar way, as we did before, appropriate sequence of points (t_x'_k)_k=1,2,… k_0,(t_x”_k)_k=1,2,… k_0⊂ I where k_0∈ satisfying suitable properties. First, we define:
	   
    ∀ k∈∖{1}, r_k:=min{ d(x_k,x_i): i=1,2…,k-1}

     and
	
    δ_x_k:=1/2min{ϵ_x_k,δ̃_x_k,d(x_k,x),d(x_k,x^*),d(x_k,y) },     k= 1
    1/2min{ϵ_x_k,δ̃_x_k,d(x_k,x),d(x_k,x^*),d(x_k,y),r_k },     k≥ 2.

  

  We proceed now to the main claim that describes the construction of the above sequences.
  
  
  	For every k∈, there exists (t_x'_i)i=1,2,…, k, (t_x”_i)i=1,2,…, k⊂ I and δ_i ∈(0,δ_x_i), ∀ i∈{1,2,… ,k} such that:
  	
  		
  (i) If k=1 it holds that: t_x<t_x'_1<t_1≤ t'_1<t_x”_1<t_2 and if k≥ 3 it holds that, t_x”_i-1<t_x'_i<t_i≤ t'_i<t_x”_i<t_i+1, ∀ i∈{2,3,…,k}
  		
  (ii) ∀ i∈{1,2,…, k},  x'_i:=γ(t_x'_i), x”_i:=γ(t_x”_i)∈∂ B_d(x_i,δ_i)⊂ U∖ M
  		
  (iii) If k=1 then γ( (t_x,t_x'_1)) ∩ M=∅ and if k≥ 2 then ∀ i∈{2,3,…,k}, γ((t_x”_i-1,t_x'_i) )∩ M=∅
  		
  (iv) For all i∈{1,2,…, k} there exists a continuous function γ̃_x'_k,x”_k:[t_x'_k,t_x”_k]→∂ B_d(x_k,δ_k) such that γ̃_x'_k,x”_k(t_x'_k)=x'_k and γ̃_x'_k,x”_k(t_x”_k)=x”_k.
  	
  
  We omit the proof of the above Claim, since is very similar with the proof of the Claim <ref>.


We remind that in a general metric space holds[Let x_0∈ X, (x_j)_j∈⊂ A, such that x_j x_0. Therefore, x_0∈A̅. Since A is closed, it follows that A̅=A and as a result x_0∈ A=iso(A). From the last, there exists δ̃_x_0>0, B_d(x_0,δ̃_x_0)∩ A={x_0}. From the convergence of (x_j)_j∈ to x_0, there exists j_0∈ such that ∀ j≥ j_0, x_j∈ B_d(x_0,δ̃_x_0)∩ A={x_0}. I.e x_j=x_0, ∀ j≥ j_0 where the last is a contradiction since x_i≠ x_l, ∀ i,l∈, i≠ l . ] the following:

Let (X,d) be a metric space, A⊂ X, closed with iso(A)=A. If (x_j)_j∈⊂ A with x_i≠ x_l, ∀ i,l∈, i≠ l, then for all x_0∈ X, x_j↛ x_0.

As a consequence from the above result, we conclude that A:={x_k: k∈} is closed in (X,d). Moreover, since A⊂γ([t_x,t_y] ), where γ([t_x,t_y] ) is compact as a continuous image of a compact set, we conclude that A is also compact as a closed subset of a compact set. Furthermore, since {x^*} is a compact set, (as a finite), from a known result in metric spaces, it follows that 

    ∃ k_0∈, dist(x^*, A):=inf{ d(x^*,s): s∈ A} =d(x^*,x_k_0).

Due to the fact that the sequence  (x_k)_k∈ consists of discrete points (i.e ∀ i,l∈, i≠ l, x_i≠ x_l) combined with (U-M) property of the set U , we conclude that there exists a continuous path γ^*_x_k_0,x^*:[t'_k_0,t^*]→ U such that, x_k_0=γ^*_x_k_0,x^*(t'_k_0) and x^*=γ^*_x_k_0,x^*(t^*), with γ^*_x_k_0,x^*(( t_x'_k_0,t^*))∩ M=∅.
 	
By the selection of δ_k_0 we obtain that δ_k_0<1/2min{ϵ_x_k_0,d(x_k_0,x_k^*)}. Then from Proposition <ref> it holds that 

    ∅≠K̃”_x_k_0:=γ^*_x_k_0,x^*([t'_k_0,t^*])∩∂ B_d(x_k_0,δ_k_0).

	Therefore, there exists, z̃”_k_0∈[t'_k_0,t^*], γ^*_x_k_0,x^*(z̃”_k_0)∈∂ B_d(x_k_0,δ_k_0).
	We define,
	
    t̃_x_k_0:=sup{ t∈[t'_k_0,t^*]: γ^*_x_k_0,x^*(t)∈∂ B_d(x_k_0,δ_k_0) }
    x̃”_k_0:=γ^*_x_k_0,x^*(t̃_x_k_0).

As we can easilly see, t̃_x_k_0 is well defined and moreover from the continuity of γ^*_x_k_0,x^* and the fact that ∂ B_d(x_k_0,δ_k_0) is closed, we obtain that x̃”_k_0∈∂ B_d(x_k_0,δ_k_0)⊂ U∖ M. Using now the information of the theorem, we obtain that the boundary of the ball B_d(x_k_0,δ_k_0) is path connected. Therefore, there exists a continuous function γ̃_x'_k_0,x̃”_k_0:[t_x'_k_0,t̃_x_k_0]→∂ B_d(x_k_0,δ_k_0)⊂ U∖ M, with x'_k_0= γ̃_x'_k_0,x̃”_k_0(t_x'_k_0) and x̃”_k_0=γ̃_x'_k_0,x̃”_k_0(t̃_x_k_0).

Due to the fact that, x^*∈ M, by the assumptions of theorem, there exist ϵ_x^*>0,δ̃_x^*>0 such that B_d(x^*,ϵ_x^*)⊂ U and B_d(x^*,δ̃_x^*)∩ M={x^*}.
We define:

    δ_x^*:=1/2min{ϵ_x^*,δ̃_x^*,d(x^*,x),d(x^*,y),d(x^*,x_k_0) }
    δ^*_x^*:=1/2min{δ_x^*,d(x^*,x̃”_k_0)}.
 
We observe that δ^*_x^*>0. Indeed, d(x^*,x̃”_k_0)>0 since,

    x̃”_k_0∈∂ B_d(x_k_0,δ_k_0)⊂B̅_d(x_k_0,δ_k_0)⊂ B_d(x_k_0,δ_x_k_0)⊂ B_d(x_k_0,1/2d(x_k_0,x^*)) and x^*∈ B_d(x^*,1/2d(x^*,x_k_0))

and B_d(x_k_0,1/2d(x_k_0,x^*))∩ B_d(x^*,1/2d(x^*,x_k_0))=∅. Therefore, d(x^*,x̃”_k_0)>0. Furthermore, we observe that 

    δ^*_x^*≤1/2min{ϵ_x^*,d(x^*,x̃”_k_0)} and δ^*_x^*≤1/2min{ϵ_x^*,d(x^*,y)}.

From the Proposition, <ref>, we receive that ∀δ∈(0,δ^*_x^*]:

    ∅≠ K'_x^*:=γ^*_x_k_0,x^*([t̃_x_k_0,t^*] ) ∩∂ B_d(x^*,δ) and ∅≠ K”_x^*:=γ(t^*,t_y )∩∂ B_d(x^*,δ)

We fix such δ^(x^*)∈ (0,δ^*_x^*). As a result from the last, 

    ∃ z'_1∈[t̃_x_k_0,t^*]: γ^*_x_k_0,x^*(z'_1)∈∂ B_d( x^*,δ^(x^*)) , ∃ z'_2∈[t^*,t_y]: γ(z”_2)∂ B_d( x^*,δ^(x^*)) .

Next, we define, 

    t_(x^*)':=inf{ t∈[t̃_x_k_0,t^*]: γ^*_x_k_0,x^*(t)∈∂ B_d( x^*,δ^(x^*)) }
    
    	t_(x^*)”:=sup{ t∈[t^*,t_y]: γ(t)∈∂ B_d( x^*,δ^(x^*))  }
    We set (x^*)':=γ^*_x_k_0,x^*( t_(x^*)') , (x^*)”:=γ( t_(x^*)”) .

From the above, we conclude that t_(x^*)', t_(x^*)” are well defined and by continuity of γ^*_x_k_0,x^*, γ and the fact that ∂ B_d( x^*,δ^(x^*)) we receive also that  (x^*)', (x^*)”∈∂ B_d( x^*,δ^(x^*)). Furthermore, γ^*_x_k_0,x^*( [t̃_x_k_0,t_(x^*)']) ∩ M=∅, since γ^*_x_k_0,x^*( [t̃_x_k_0,t_(x^*)']) ∩ M⊂γ^*_x_k_0,x^*( [t̃'_k_0,t^*]) ∩ M=∅ where the last equality holds by the construction property of γ^*_x_k_0,x^*. Moreover, γ( [t_(x^*)”,t_y])∩ M=∅, since γ( [t_(x^*)”,t_y])∩ M⊂γ( (t^*,t_y))∩ M=∅ where the last holds from Claim <ref> (iii).




Finally, we define, for the case where k^* =∞, the path γ̂:[t_x,t_y]→ U∖ M as follows:

    γ̂(t):=γ(t),     t∈[t_x,t_x'_1]
    γ̃_x'_1,x”_1(t),     t∈[t_x'_1,t_x”_1]
    γ(t),     t∈[t_x”_1,t_x'_2]
    ⋮    ⋮
    γ(t),     t∈[t_x”_k_0-1,t_x'_k_0]
    γ̃_x'_k_0,x̃”_k_0(t),     t∈[t_x'_k_0,t̃_x_k_0]
    γ^*_x_k_0,x^*(t),     t∈[t̃_x_k_0,t_(x^*)']
    γ̃_(x^*)',(x^*)”(t),     t∈[t_(x^*)',t_(x^*)”]
    γ(t),     t∈[t_(x^*)”,t_y].



	
	
	
	
	
	
	
	
	
	
	
	


	
	
	
	
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		Savvas Andronicou
 University of Cyprus 
 Department of Mathematics & Statistics 
 P.O. Box 20537

		Nicosia, CY- 1678 CYPRUS
		
 andronikou.savvas@ucy.ac.cy
	
	
	
	
	
	
	
	
		Emmanouil Milakis
 University of Cyprus 
 Department of Mathematics & Statistics 
 P.O. Box 20537

		Nicosia, CY- 1678 CYPRUS
		
 emilakis@ucy.ac.cy
	
	
