A combinatorial characterization of Hurewicz cofibrations between finite topological spaces
Nicolás Cianci Miguel Ottina

Cofibrations between finite topological spaces]A combinatorial characterization of Hurewicz cofibrations between finite topological spaces


Facultad de Ciencias Exactas y Naturales 
 Universidad Nacional de Cuyo and CONICET 
 Mendoza, Argentina.
nicocian@gmail.com


Facultad de Ciencias Exactas y Naturales 
 Universidad Nacional de Cuyo 
 Mendoza, Argentina.
mottina@fcen.uncu.edu.ar


[2010]Primary: 06A07; 55P05. Secondary: 54C15.



Research partially supported by grants M015 (2013–2015) and M044 (2016–2018) of SeCTyP, UNCuyo. The first author was also partially supported by a CONICET doctoral fellowship.


We characterize the Hurewicz cofibrations between finite topological spaces, that is, the continuous functions between finite topological spaces that have the homotopy extension property with respect to all topological spaces. In particular, we show that cofibrations between connected non-empty finite topological spaces are homotopy equivalences.

As a consequence of our characterization, we obtain a simple algorithm capable of determining whether a given continuous function between finite topological spaces is a cofibration.


[
    Miguel Ottina
    December 30, 2023
=====================





§ INTRODUCTION


Finite topological spaces are naturally endowed with a very interesting combinatorial flavour that is based in the well-known bijective correspondence between topologies in a finite set X and preorders in X given by Alexandroff <cit.>. Under this bijection T_0 topologies correspond to partial orders. Moreover, any finite T_0–space is weak homotopy equivalent to the geometric realization of the order complex of its associated poset <cit.>. These results permit the study of homotopy properties of the order complex of a finite poset by means of its associated finite T_0–space, which has proved to be very fruitful <cit.>. Indeed, the theory of finite topological spaces gives interesting tools to study posets and polyhedra.

A natural question to pose in the homotopy theory of finite spaces is which maps between finite spaces are Hurewicz cofibrations. For example, it is easy to prove that if ={0,1} is the Sierpinski space (where {0} is the only non-trivial open set of ), then the inclusion {0}↪ is a (Hurewicz) cofibration but the inclusion {1}↪ is not. The main result of this article can be seen as both an explanation and an extensive generalization of this fact, and allows us to completely characterize the cofibrations between finite (not necessarily T_0) spaces. In particular, we obtain the unexpected results that cofibrations between connected non-empty finite spaces are homotopy equivalences and that closed cofibrations between connected non-empty finite spaces are homeomorphisms.

Our characterization of cofibrations between finite spaces is purely combinatorial and surprisingly simple: if X is a connected finite topological space and A⊆ X is a non-empty subspace then the inclusion map i A↪ X is a cofibration if and only if there exists a retraction r X → A of i such that ir≤𝕀_X. Moreover, we prove that for finite T_0–spaces such a retraction exists if and only if the subspace A is, in our terminology, a dbp–retract of X, that is, the subspace A can be obtained from X by successively removing down beat points (which are points that have exactly one lower cover). In addition, we prove that a inclusion map i A↪ X as above is a cofibration if and only if the induced map between the Kolmogorov quotients of A and X is a cofibration. As a corollary of these results we obtain a simple algorithm for determining whether a continuous map between finite topological spaces is a cofibration.

Dbp–retracts turn out to have very interesting properties which are studied in section <ref> of this article. For example, for any finite T_0–space X, the set Ω(X,∅) of dbp–retracts of X is in one-to-one correspondence with the set 

	(X,∅)={f∈X^X/f≤𝕀_X and f^2=f},

where X^X is the space of continuous functions from X to itself.

The set (X,∅) is not closed under compositions, but is closed under the operation * defined by f*g=(fg)^∞, where f^∞ denotes the composition of f with itself a sufficiently large amount of times. It is not hard to see that * turns (X,∅) into an abelian monoid, which is a consequence of the fact that the set Ω(X,∅) is closed under intersections. In particular, every finite T_0–space X has a unique minimal dbp–retract. This means that successively removing down beat points of X furnishes a space without down beat points which is independent of the order in which the down beat points of X (and the successively obtained subspaces) are removed.



§ PRELIMINARIES

For every finite topological space X and for every x∈ X, there is a minimal open set that contains x, which will be denoted by U^X_x (or by U_x, if the space X is understood). The space U^X_x-{x} will be denoted by U^X_x.

If X is a finite topological space, a preorder ≤ is defined on X as follows: for x,y∈ X,

	 x≤y  if and only if  U_x⊆U_y.   

Note that ≤ is nothing but the specialization preorder in X, that is, x≤ y if and only if y is in the closure of {x}. Thus, ≤ is an order if and only if X is a T_0–space.
On the other hand, if X is a finite preordered set, the lower sets of X form a topology on X. It is well known that these constructions are mutually inverse and provide a functorial bijective correspondence between finite spaces and finite preordered sets that restricts to a bijective correspondence between finite T_0–spaces and finite posets <cit.>.  
Hence, from now on, every finite space will be considered as a finite preordered set, every finite T_0–space will be considered as a finite poset and every continuous function between finite spaces will be considered as an morphism of preordered sets without further notice. 

Let X be a finite space considered as a preordered set with preorder ≤. The preorder ≤^ is defined on X by 
	
		x≤^ y  if and only if  y≤x
	
for x,y∈ X. The preorder ≤^ induces a topological space X^ with the same underlying set as X but whose open sets are the closed sets of X. A continuous function f X→ Y between finite spaces can be regarded as  a continuous function f^ X^→ Y^.  

Now, if X is a finite space, an equivalence relation ∼ on X is defined by

	 x∼y  if and only if  U_x=U_y.   

Hence, x∼ y if and only if x≤ y and y≤ x. We will denote X/∼ by X_0 and the canonical quotient map X→ X_0 by q_X. Note that every function j X_0→ X such that j(q_X(x))∼ x for all x∈ X is continuous (since jq_X is order preserving) and hence a section of q_X. 
It is easy to see that X_0 is a T_0–space and that x≤ x' in X if and only if q_X(x)≤ q_X(x') in X_0 <cit.>.

Observe that a continuous function f X→ Y induces a unique continuous function f_0 X_0→ Y_0 such that f_0q_X=q_Yf. The assignments X↦ X_0 and f↦ f_0 define a functor T from the category  of finite topological spaces (and continuous functions) to the category  of finite T_0–spaces (and continuous functions). The functor T is left-adjoint to the inclusion functor i→.

For a finite T_0–space X and elements x,y in X, we will write x<y when x≤ y and x≠ y. We will also write x≥ y if y≤ x and x>y if y<x.

If X and Y are finite spaces, the space of continuous functions from X to Y (equipped with the compact-open topology) will be denoted by Y^X. 
Note that, in this case, Y^X is a finite topological space, and for any two continuous functions f,g∈ Y^X we have that f≤ g in Y^X if and only if f(x)≤ g(x) for every x in X <cit.>.
Thus, for every f,g∈ Y^X, we have that f≤ g in Y^X if and only if f_0≤ g_0 in Y_0^X_0.

In <cit.>, R. E. Stong also proved that a finite space is connected if and only if it is connected when considered as a preordered set, and if and only if it is path-connected. In the same work, it is proved that if X and Y are finite topological spaces and f,g∈ Y^X are such that f≤ g, then f is homotopic to g relative to the set {x∈ X:f(x)=g(x)}.

Let X be a finite T_0–space and let x∈ X. As in <cit.>, we say that x is a down beat point of X if the set U^X_x={z∈ X:z<x} has a maximum, or equivalently, if the element x has exactly one lower cover.

Similarly, we say that x is an up beat point of X if the set {z∈ X:z>x} has a minimum.

We say that x is a beat point of X if it is either a down beat point of X or an up beat point of X.


	Let X be a finite T_0–space and let x be a down beat point of X. Let i X-{x}→ X be the inclusion and let r X→ X-{x} be the function defined by
	
		r(z)=z   if z≠ x,
max(U_x)   if z=x.
	
	It is easy to see that r is continuous, that ri=𝕀_X-{x} and that ir≤𝕀_X. In particular, X-{x} is a strong deformation retract of X.
	


We will need the following well-known result, which is easy to prove. 

	Let X be a connected finite space, let Y be a T_1–space and let f X→ Y be a continuous function. Then f is a constant map.
	




§ BP–RETRACTS OF FINITE T_0–SPACES



In this section, we introduce the concepts of dbp–retracts and ubp–retracts of finite T_0–spaces and prove some properties that will be needed in section <ref>.  

	Let X be a finite T_0–space and let A⊆ X. We will say that A is a dbp–retract (resp. ubp–retract) of X if A can be obtained from X by successively removing down beat points (resp. up beat points), that is, if there exist n∈_0 and a sequence X=X_0⊇ X_1⊇⋯⊇ X_n=A of subspaces of X such that, for all i∈{1,…,n}, the space X_i is obtained from X_i-1 by removing a single down beat point (resp. up beat point) of X_i-1.
	
	We will say that A is a bp–retract of X if A is either a dbp–retract or a ubp–retract of X.


In particular, X is a dbp–retract and a ubp–retract of itself.


	
	If Y is a finite T_0–space, X is a dbp–retract of Y and A is a dbp–retract of X, then A is a dbp–retract of Y.



	
	Note that if X is a finite T_0–space and A is a dbp–retract of X then the minimal points of X are contained in A, since they cannot be down beat points of any subspace of X. In particular, A must be dense in X. 


	
	If X is a finite T_0–space, then a subspace A of X is a ubp–retract of X if and only if A^ is a dbp–retract of X^.

	In the rest of this section, we will prove several results for dbp–retracts of finite T_0–spaces. Similar results hold for ubp–retracts by <ref>.
	
 
	Let X be a finite T_0–space, let A be a subspace of X and let i A→ X be the inclusion map. Then, the following propositions are equivalent:
	
		
  * A is a dbp–retract of X.
		
  * There exists a continuous function f X→ X such that f≤𝕀_X, f^2=f and f(X)=A.
		
  * There exists a unique continuous function f X→ X such that f≤𝕀_X, f^2=f and f(X)=A.
		
  * There exists a retraction r X→ A of i such that ir≤𝕀_X.
		
  * There exists a unique retraction r X→ A of i such that ir≤𝕀_X.
	

 We will prove that (1)⇒ (4)⇒ (5)⇒ (3)⇒ (2)⇒ (1).

	The implication (1)⇒ (4) follows from <ref>.
	
	Next, we will show that (4)⇒ (5).
	For k=1,2, let r_k X→ A be a retraction of i such that ir_k≤𝕀_X. We wish to show that that r_1=r_2. Let x∈ X. 
	Since r_2(x)∈ A and r_2(x)≤ x it follows that r_2(x)=r_1r_2(x)≤ r_1(x). Similarly, r_1(x)≤ r_2(x) and thus, r_1(x)=r_2(x). Therefore, r_1=r_2.
	
	We will now prove that (5)⇒ (3). Suppose that there exists a unique retraction r X→ A of i such that ir≤𝕀_X. It is clear that (ir)^2=ir and that ir(X)=A. Thus, we have proved that (5) implies (2).
	
	Now, for k=1,2, let f_k X→ X be a continuous function such that f_k≤𝕀_X, f_k^2=f_k and f_k(X)=A and let r_k X→ A be the range restriction of f_k. Since f_k^2=f_k and f_k≤𝕀_X, it is clear that r_k is a retraction of i such that ir_k≤𝕀_X for k=1,2. Hence, r_1=r_2. It follows that f_1=f_2.
	
	The implication (3)⇒ (2) is clear. 
	
	Next, we will show that (2)⇒ (1). Let f X→ X be a continuous function such that f≤𝕀_X, f^2=f and f(X)=A.	
	
	Let W={x∈ X:f(x)<x}. If W=∅ then X=f(X)=A and the result follows. 
	
	Suppose that W≠∅.
	Let x_0 be a minimal element of W. Since x_0∈ W, f(x_0)<x_0. We claim that f(x_0)=maxU_x_0. Indeed, if x_1<x_0 then x_1∉W and therefore x_1=f(x_1)≤ f(x_0). It follows that x_0 is a down beat point of X.
	
	Let X'=X-{x_0}. Thus, X' is obtained from X by removing a single down beat point. Since f∘ f=f it follows that x_0∉f(X), and therefore, we can restrict f to a function f' X'→ X'. It is clear that f'≤𝕀_X' and that f'∘ f'=f'. Moreover, f'= f=A since f(x_0)=f(f(x_0))∈ f(X'). The result follows by an inductive argument.


	
	Let X be a finite T_0–space, let A be a dbp–retract of X and let i A→ X be the inclusion map. From the proof of <ref>, it is clear that the unique continuous function f X→ X such that f≤𝕀_X, f^2=f and f(X)=A and the unique retraction r X→ A of i such that ir≤𝕀_X are related by f=ir. Equivalently, r is the range restriction of f to its image A.


	
	Let X be a finite T_0–space and let A be a subspace of X. Then, A is a dbp–retract of X if and only if U^X_x∩ A has a maximum for every x∈ X. Equivalently, A is a dbp–retract of X if and only if U^X_x∩ A has a maximum for every x∈ X-A.


	Suppose that A is a dbp–retract of X. Let r X→ A be the only retraction of i such that ir≤𝕀_X and let x∈ X. It is clear that r(x)∈ U^X_x∩ A. Now, if y∈ U^X_x∩ A then y=r(y)≤ r(x) since y≤ x. Therefore, r(x) is the maximum of U^X_x∩ A.
	
	Now, suppose that U^X_x∩ A has a maximum for every x∈ X-A.
	Note that a is the maximum of U^X_a∩ A for every a∈ A. 
	Let r X→ A be defined by r(x)=max (U^X_x ∩ A). If x≤ x' then U^X_x⊆ U^X_x' and hence r(x)≤ r(x'). It follows that r is continuous. It is clear that ri=𝕀_A and that ir≤𝕀_X. Hence, by <ref> A is a dbp–retract of X.


	Let Y be a finite T_0–space, let X⊆ Y and let A⊆ X be a dbp–retract of Y. Then A is a dbp–retract of X.
	


	Let i A→ X and j X→ Y be the inclusion maps and let r Y→ A be a continuous function such that rji=𝕀_A and jir≤𝕀_Y. Then rj X→ A satisfies rji=𝕀_A and irj≤𝕀_X, and thus, A is a dbp–retract of X by <ref>.


	Let Y be a finite T_0–space, let X be a dbp–retract of Y and let A⊆ X. Then A is a dbp–retract of X if and only if A is a dbp–retract of Y.
	


	Immediate from 	<ref> and <ref>.


	Let X be a finite T_0–space and let A_1 and A_2 be two dbp–retracts of X. For k=1,2, let i_k A_k→ X be the inclusion and let r_k X→ A_k be the unique retraction of i_k such that i_k r_k≤𝕀_X. Then A_1⊆ A_2 if and only if i_1 r_1≤ i_2 r_2.
	


	Suppose that A_1⊆ A_2 and let i A_1→ A_2 be the inclusion. By <ref>, A_1 is a dbp–retract of A_2. Hence, there exists a retraction r of i such that ir≤𝕀_A_2. Now, since i_2 i=i_1, it follows that rr_2i_1=rr_2i_2i=𝕀_A_1 and i_1rr_2=i_2irr_2≤ i_2r_2≤𝕀_X. By <ref>, rr_2=r_1. Thus, i_1r_1=i_2irr_2≤ i_2r_2.
	
	Now, suppose that i_1r_1≤ i_2r_2 and let w∈ A_1. Then w=i_1r_1(w)≤ i_2r_2(w)≤ w and it follows that i_2r_2(w)=w. Hence, w∈ A_2. The result follows.


	
	Let X be a finite T_0–space and, for k=1,2, let f_k X→ X be a continuous function such that f_k∘ f_k=f_k and f_k≤𝕀_X. Then f_1≤ f_2 if and only if f_1(X)⊆ f_2(X).


	By <ref>, f_1(X) and f_2(X) are dbp–retracts of X. The result follows from <ref> and <ref>.



	Let X be a finite T_0–space and let A be a subspace of X. We define
		
		(X,A)={f∈X^X:f≤𝕀_X, f^2=f, and A⊆f(X)}
	
	and
	
		Ω(X,A)={W⊆X:W is a dbp–retract of X and A⊆ W}.
	
	
	The set (X,A) will be considered as a subposet of X^X and the set Ω(X,A) will be considered as a poset with the order given by set inclusion. 


	Let X be a finite T_0–space and let X_m denote the set of minimal elements of X. By <ref>, it is clear that Ω(X,∅)=Ω(X,A) for every A⊆ X_m.




	Let X be a finite T_0–space and let A be a subspace of X. Then (X,A) is order isomorphic to Ω(X,A).
	


	By <ref>, there is a bijection φ(X,A)→Ω(X,A) defined by φ(f)=f(X) for every f∈(X,A). By <ref>, φ and its inverse are order-preserving functions. The result follows.


	Let X be a finite T_0–space and let f X→ X be a continuous function such that f≤𝕀_X. Since f≥ f^2≥ f^3≥… and X^X is finite, there exists N∈ such that f^N+1=f^N. It is clear that f^n=f^N for every n≥ N. This motivates the following definition.

	Let X be a finite T_0–space and let f X→ X be a continuous function such that f≤𝕀_X. We define f^∞ by f^N where N∈ is such that f^N=f^N+1. 


The following lemma states some simple properties of the construction of the previous definition.

 
Let X be a finite T_0–space and let f,g X→ X be continuous functions such that f≤𝕀_X and g≤𝕀_X. Then:


  * f^∞≤ f≤𝕀_X.

  * f^∞∘ f^∞=f^∞.

  * f^∞(X) is a dbp–retract of X.

  * For all x∈ X, x∈ f^∞(X) if and only if f(x)=x.

  * If f≤ g then f^∞≤ g^∞.

  * (fg)^∞(X)=f^∞(X)∩ g^∞(X).




The first two items follow easily from the definition of f^∞. The third item follows from items (1) and (2) and theorem <ref>. The proof of items (4) and (5) are easy and will be omitted.

Now we will prove (6). Since fg≤ f, (fg)^∞≤ f^∞. Thus, (fg)^∞(X)⊆ f^∞(X) by <ref>. Similarly, (fg)^∞(X)⊆ g^∞(X). Hence (fg)^∞(X)⊆ f^∞(X) ∩ g^∞(X). Now, if x∈ f^∞(X) ∩ g^∞(X) then f(x)=x=g(x) by item (4). Thus, x=(fg)^∞(x)∈ (fg)^∞(X).



	
	Let X be a finite T_0–space and let A⊆ X.
	
	Then, for every f,g∈(X,A):
	
		
  * (fg)^∞∈(X,A), and
		
  * (fg)^∞(X)=f(X)∩ g(X).
	



Let f,g∈(X,A). Note that f^∞=f and g^∞=g. Thus

A ⊆f(X) ∩g(X) = f^∞(X) ∩g^∞(X) = (fg)^∞(X)

by item (6) of <ref>. The result follows from items (1) and (2) of <ref>.



	Let X be a finite T_0–space and let A⊆ X.
		
	Then, Ω(X,A) is closed under intersections. In particular, Ω(X,A) has a minimum.
	


	Let W_1,W_2∈Ω(X,A). Let φ be as in the proof of <ref> and let f_k=φ^-1(W_k) for k=1,2. By <ref>, we have that W_1∩ W_2=f_1(X)∩ f_2(X)=(f_1f_2)^∞(X)∈Ω(X,A). 


	Let X be a finite T_0–space and let A and B be two dbp–retracts of X. Then A∩ B is a dbp–retract of X. 


	
	Let X be a finite T_0–space and let A be a subspace of X. Proposition <ref> implies that A is a dbp–retract of X if and only if A is the minimum element of Ω(X,A). From <ref> it follows that A is a dbp–retract of X if and only if any sequence of successive removals of down beat points of X which do not belong to A ends with the subspace A when all such down beat points have been removed. Therefore, this gives an efficient algorithm to decide whether a subspace of a finite T_0–space X is a dbp–retract of X.
	
	Observe that Theorem <ref> gives another efficient algorithm for doing the same. These two algorithms are esentially equivalent since after considering a linear extension of the partial order of X-A, the algorithm given by <ref> is equivalent to the identification (and removal) of down beat points in increasing order.



 
Let X={a,b,c,d,e,f,g,h} be the T_0–space that corresponds to the following Hasse diagram:

[x=2cm,y=2cm]
every node=[font=]
(1,0) node(a)∙ node  [below] a;
(2,0) node(b)∙ node [below]b;
(0,1) node(c)∙ node [left]c;
(1,1) node(d)∙ node  [left] d;
(2,1) node(e)∙ node  [right] e;
(3,1) node(f)∙ node [right]f;
(1,2) node(g)∙ node [above]g;
(2,2) node(h)∙ node [above]h;
(g) – (c);
(g) – (e);
(g) – (f);
(h) – (d);
(h) – (e);
(h) – (f);
(c) – (a);
(c) – (b);
(d) – (a);
(d) – (b);
(e) – (a);
(f) – (b);



Let A={a,b,c,d} and let B={a,b,d,g}, both of them considered as subspaces of X.
We can obtain the subspace A from X by successively removing the down beat points e, f, g and h.

Now, we can obtain B by successively removing the beat points e, f, h and c. Thus, B is a strong deformation retract of X. However,
the set {a,b,c,d,g} is minimal, with respect to set inclusion, among the dbp–retracts of X that contain B. This dbp–retract is not equal to B, and hence B is not a dbp–retract of X.
Observe that the same conclusion is achieved applying <ref> and noting that the set U^X_c∩ B does not have a maximum element. 





§ COFIBRATIONS BETWEEN FINITE TOPOLOGICAL SPACES



By cofibration we will mean Hurewicz cofibration, that is, a continuous function which has the homotopy extension property with respect to all topological spaces.

It is well known that every cofibration is a homeomorphism onto its image <cit.>.

A classic result states that if X is a topological space and A is a subspace of X such that the inclusion of A into X is a cofibration, then X×{0}∪ A× I is a retract of X× I. The converse is easy to prove when A is a closed subspace of X and was proved without this assumption by Strøm in <cit.> using the following lemma.


	
	Let X be a topological space and let A be a subspace of X such that X×{0}∪ A× I is a retract of X× I. Then, a subset C of X×{0}∪ A× I is open if and only if C∩ X×{0} is open in X×{0} and C∩ A× I is open in A× I.

From this lemma, he obtained the following characterization of cofibrations.
[Strøm, <cit.>]
	
	Let X be a topological space and let A be a subspace of X. Then, the inclusion i A↪ X is a cofibration if and only if X×{0}∪ A× I is a retract of X× I.


In this section, we will obtain a simple and combinatorial characterization of cofibrations between finite topological spaces and we will show how it is related to the notion of dbp–retracts of section <ref>. To this end, we will give a simple alternative proof of Lemma <ref> in the case that the subspace A is a finite space (Lemma <ref>) from which our characterization of cofibrations between finite spaces will be obtained. 

Our proof of lemma <ref> is based on Strøm's proof of <ref>. Nevertheless, it is interesting to observe that, under the assumption that A is a finite space, the hypothesis that X×{0}∪ A× I is a retract of X× I of Lemma <ref> is not required for Lemma <ref>. Moreover, by <ref>, this hypothesis holds if and only if the inclusion map A↪ X is a cofibration. Thus, from the characterization of cofibrations between finite spaces that will be given in this section, one can construct many examples of inclusion maps A↪ X for which <ref> can be applied but <ref> can not.


 
	Let X be a topological space, let A be a finite subspace of X. Then, a subset C of X×{0}∪ A× I is open in X×{0}∪ A× I if and only if C∩ X×{0} is open in X×{0} and C∩ A× I is open in A× I.


	Suppose that C∩ X×{0} is open in X×{0} and C∩ A× I is open in A× I. Let Y=X×{0}∪ A× I. 
	Let U={x∈ X:(x,0)∈ C}. It is clear that U is open in X. Since U∩ A is finite and C∩ A× I is open in A× I, then there exists ε>0 such that (U∩ A)× [0,ε)⊆ C.
	
	It is easy to see that C=(U× [0,ε)∩ Y)∪(C∩ A× (0,1]).
	
	Furthermore, U× [0,ε)∩ Y is clearly open in Y. On the other hand, since C∩ (A×(0,1]) is open in A× (0,1] and A× (0,1] is open in Y, it follows that C∩(A× (0,1]) is open in Y. Thus, C is open in Y.
	
	The converse is clear.


The following proposition follows easily from the previous lemma.


	Let X and Z be topological spaces and let A be a finite subspace of X. Let f X→ Z and H A× I → Z be two continuous functions such that H(a,0)=f(a) for every a∈ A. Then, the function ϕ X×{0}∪ A× I→ Z defined by
	
		ϕ(x,t)=f(x)    if t=0,
H(x,t)    if x∈ A
	
	is continuous.
	
	Equivalently, X×{0}∪ A× I is the mapping cylinder of the inclusion map A↪ X.


From the previous proposition we obtain the following corollary which is a particular case of <ref> but was obtained with a much simpler proof.


	
	Let X be a topological space, let A be a finite subspace of X and let i X×{0}∪ A× I→ X× I be the inclusion map. Then the inclusion A↪ X is a cofibration if and only if there exists a retraction r of i. 


The following is one of the main results of this article.

	Let X be a connected finite topological space and let A be a non-empty subspace of X. Then, the inclusion i A↪ X is a cofibration if and only if there exists a retraction r X→ A of i such that ir≤𝕀_X. 
	


	Let Y=X×{0}∪ A× I and let ι Y→ X× I be the inclusion. We will prove that ι has a retraction if and only if i has a retraction r such that ir≤𝕀_X. Hence, the theorem will follow from <ref> (or <ref>).
	
	Suppose that ι has a retraction ρ. Let p_X X× I→ X and p_I X× I→ I the canonical projections. 
	For each t∈ I, we have a continuous function i_t X→ X× I defined by i_t(x)=(x,t) for every x∈ X. Note that p_I ιρ i_t is a continuous function from X to I, and hence, it is a constant map by <ref>. Now, if a∈ A, p_I ιρ i_t(a)=p_I ιρ(a,t)=p_I(a,t)=t. It follows that p_I ιρ i_t(x)=t for every x∈ X.
	
	Let ϕ I→ X^X be the function induced by the map p_Xιρ X× I → X. Note that ϕ is continuous by <cit.> and that ϕ(0)=𝕀_X. Hence, 0∈ϕ^-1(U_𝕀_X) and thus, there exists ε>0 such that ϕ(ε)≤𝕀_X. It can be readily verified that ϕ(ε)(x)∈ A for every x∈ X. Hence, we can restrict ϕ(ε) to a map  r X→ A, and it is clear that ir=ϕ(ε)≤𝕀_X. And since ιρ is the identity on A× I, it easily follows that ri=𝕀_A.

	For the converse, suppose that there is a retraction r of i such that ir≤𝕀_X. Let α I→ X^X be defined by
	
	α(t)=𝕀_X   if t=0,
ir    if t>0.
	
	Since ir≤𝕀_X, it follows that α is a continuous map. Let α^♭ X× I → X be the function induced by α and the exponential law. We obtain that α^♭ is continuous by <cit.>.
	
	As above, let p_I X× I→ I be the canonical projection. Let β X× I→ X× I be the map induced by α^♭ and p_I. It is easy to verify that β⊆ Y. Let ρ X× I→ Y be the range restriction of β to Y. It is not difficult to check that ρ is a retraction of ι.


 
Let X be a finite topological space and let A be a subspace of X. Observe that, in order to determine whether the inclusion map A↪ X is a cofibration, we may always reduce our analysis to cases in which the hypotheses of the previous theorem are fulfilled. Indeed, since a finite topological space is the coproduct of its connected components, it follows that the inclusion map A↪ X is a cofibration if and only if the inclusion map A∩ C ↪ C is a cofibration for each connected component C of X. In addition, since the inclusion of the empty subspace in any topological space is a cofibration, we obtain that the inclusion map A↪ X is a cofibration if and only if the inclusion map A∩ C ↪ C is a cofibration for each connected component C of X such that C∩ A≠∅.


Theorem <ref> yields the unexpected results <ref>, <ref> and <ref>.

 
	Let X be a connected finite space and let A be a non-empty subspace of X. If the inclusion i A→ X is a cofibration, then A is a strong deformation retract of X.

 
	Let X be a connected finite space, let A be a non-empty closed subspace of X. If the inclusion i A→ X is a cofibration, then A=X.


	Suppose that i is a cofibration. By <ref>, there exists r X→ A such that ri=𝕀_A and ir≤𝕀_X.
	Now let x∈ X. We have that x≥ r(x)∈ A. Since A is closed, x∈ A. The result follows. 

 
	A closed cofibration between non-empty connected finite spaces is a homeomorphism.

Recall that a pointed space (X,x_0) is said to be well-pointed if the inclusion {x_0}↪ X is a cofibration.  

	Let (X,x_0) be a pointed connected finite space. Then (X,x_0) is well-pointed if and only if x_0≤ x for every x∈ X. In particular, if X is a T_0 space, (X,x_0) is well-pointed if and only if x_0 is the minimum of X.


	By <ref>, the space (X,x_0) is well-pointed if and only if the only map r X→{x_0} satisfies that r(x)≤ x for every x∈ X.


In the case of connected finite T_0–spaces, non-trivial cofibrations are essentially dbp–retracts as the following result and its corollaries state.


	Let X be a connected finite T_0–space and let A be a non-empty subset of X. Then, the inclusion i A→ X is a cofibration if and only if A is a dbp–retract of X.
	



	Immediate from <ref> and <ref>.



	
	Let X and Y be finite T_0–spaces such that Y is connected and X≠∅. Let f X→ Y be any function. Then f is a cofibration if and only if f is a homeomorphism onto its image and f(X) is a dbp–retract of Y.


	Let X be a connected finite T_0–space. Then, every cofibration f A→ X with A≠∅ is a homeomorphism if and only if X does not have down beat points. 



We will prove now that in order to determine if a map between finite spaces is a cofibration we can always reduce our analysis to maps between finite T_0–spaces. To this end, we need some simple results which are contained in the following remark.

 
 Let X be a finite topological space and let q_X X→ X_0 be the quotient map.

 
  * The set of connected components of X_0 is {q_X(C) C  is a connected component of X}.
 
  * Let A be a subspace of X and let i A→ X be the inclusion map. Then the map i_0 A_0→ X_0 is an embedding and A_0 is canonically homeomorphic to q_X(A). In addition, q_X(A∩ C)=q_X(A)∩ q_X(C) for each connected component C of X.



The following proposition allows the reduction to the case of finite T_0–spaces in the characterization of cofibrations.

 
Let X be a finite topological space and let A be a subspace of X. Then, the inclusion i A→ X is a cofibration if and only if the map i_0 A_0→ X_0 is a cofibration.



It is easy to prove that the map i_0 A_0→ X_0 is a retract of the inclusion i A→ X. Thus, if i is a cofibration then i_0 is also a cofibration.


For the converse, suppose first that X is connected and A≠∅.

Let q_A A→ A_0 and q_X X→ X_0 be the canonical quotient maps and let j_A A_0→ A be a section of q_A. 
By <ref>, there exists a retraction ρ of i_0 such that i_0ρ≤𝕀_X_0. We define r X→ A by

r(x)=x   if x∈ A,
j_Aρq_X(x)   if x∈ X-A.


Observe that j_Aρ q_X(a)∼ a for every a∈ A. It follows that 

	r(a)=a≤j_Aρq_X(a)≤j_Aρq_X(x)=r(x)

for every a∈ A and every x∈ X-A such that a≤ x, and that 

r(x)=j_Aρq_X(x)≤j_Aρq_X(a)≤a=r(a)

for every x∈ X-A and every a∈ A such that x≤ a. Hence, r is continuous.

It is clear that ri=𝕀_A. 
On the other hand, ir(x)=x for every x∈ A and, since q_Xi=i_0q_A, then

q_X ir(x)=q_Xij_Aρq_X(x)=i_0q_Aj_Aρq_X(x)=i_0ρq_X(x)≤q_X(x)

for every x∈ X-A. Hence, ir≤𝕀_X. Then, i is a cofibration by <ref>.

The general case follows applying <ref> and <ref>.


The following result follows easily from <ref>


Let X and Y be finite topological spaces and let f X→ Y be a continuous map. Then f is a cofibration if and only if f is an embedding and f_0 X_0→ Y_0 is a cofibration.


Note that a map f X→ Y between finite spaces might not be a cofibration even if f_0 X_0→ Y_0 is a cofibration. For example, let X={0,1} with the indiscrete topology, let Y be the singleton and let f X→ Y be the only possible map. Then the map f X→ Y is not a cofibration since it is not injective but the map f_0 X_0→ Y_0 is a homeomorphism.


Combining some of the results developed above we obtain a simple algorithm for determining whether a function between finite topological spaces is a cofibration, which is described below.

Let X and Y be finite topological spaces and let f X→ Y be a function. Clearly f is a cofibration if and only if the range restriction f|^f(X) X→ f(X) is a homeomorphism and the inclusion map f(X)↪ Y is a cofibration. Observe that f|^f(X) is a homeomorphism if and only if for all x_1,x_2∈ X, x_1≤ x_2 ⇔ f(x_1)≤ f(x_2).

Let A=f(X). By <ref> and <ref> the inclusion map A↪ Y is a cofibration if and only if the inclusion map q_Y(A)↪ Y_0 is a cofibration. By <ref>, this holds if and only if the inclusion map q_Y(A)∩ C ↪ C is a cofibration for each connected component C of Y_0 such that q_Y(A)∩ C≠∅. Now, for each connected component C of Y_0 such that q_Y(A)∩ C≠∅, the inclusion map q_Y(A)∩ C ↪ C is a cofibration if and only if q_Y(A)∩ C is a dbp–retract of C by <ref>.

Therefore, we obtain that the inclusion map A↪ Y is a cofibration if and only if q_Y(A)∩ C is a dbp–retract of C for each connected component C of Y_0 such that q_Y(A)∩ C≠∅. This condition can be verified algorithmically, as was noted in <ref>.



As an example of application of the previous results, we will determine if certain inclusion maps regarding mapping cylinders are cofibrations. Recall that if X and Y are topological spaces and f X→ Y is a continuous map then the inclusion maps of X and Y into the mapping cylinder of f are cofibrations.

Since the mapping cylinder of a continuous function between (non-empty) finite topological spaces is not finite, we can not apply our results to the standard mapping cylinder. However, we are interested in the discrete analog of the mapping cylinder for continuous maps between finite T_0–spaces which is more suitable for working in the finite setting (see <cit.>).


	Let X and Y be finite T_0–spaces and let f X→ Y be a continuous function. The non-Hausdorff mapping cylinder of f is the space B(f) whose underlying set is X∐ Y and whose topology is induced by the following order in B(f):
	
		z≤z' in B(f) if and only if 
		
			z≤z' in X    if z,z'∈ X,

			f(z)≤z' in Y    if z∈ X and z'∈ Y, or

			z≤z' in Y    if z,z'∈ Y.
		  
	
	The canonical inclusion maps of X and Y into B(f) will be denoted by j_X and j_Y, respectively.


	Let f X→ Y be a continuous function between non-empty finite T_0–spaces. Since Y is closed in B(f), from <ref> it follows that j_Y is not a cofibration.



	Let X and Y be finite T_0–spaces and let f X→ Y be a continuous function. Then j_Y^ is a cofibration.



	As in the proof of Lemma 2.8.2 of <cit.>,
	let j=j_Y Y→ B(f) be the inclusion map and let r B(f)→ Y defined by 
	
		r(z)=f(z)    if z∈ X,
z    if z∈ Y. 
	
	If x∈ X and y∈ Y are such that x≤ y in B(f), then f(x)≤ y and hence r(x)=f(x)≤ y=r(y). Therefore, r is continuous.
	It is clear that rj=𝕀_Y.
	
	Now, jr(x)=f(x)≥ x in B(f) for every x∈ X. Thus, jr≥𝕀_B(f). The result follows from <ref>.


The following result is already present in the proof of Proposition 4.6.6 of <cit.> with a different terminology. We give here a proof of it using our tools.

 
	Let X and Y be finite T_0–spaces and let f X→ Y be a continuous function. Then X is a dbp–retract of B(f) if and only if f^-1(U^Y_y) has a maximum for every y∈ Y.


Note that f^-1(U^Y_y)=U^B(f)_y∩ X for every y∈ Y. The result follows from <ref>.



Let X and Y be finite T_0–spaces such that Y is connected and X≠∅. Let f X→ Y be a continuous map. Then the inclusion map j_X X → B(f) is a cofibration if and only if f^-1(U^Y_y) has a maximum for every y∈ Y.



By <ref>, the map j_X X → B(f) is a cofibration if and only if X is a dbp–retract of B(f). The result follows from <ref>.




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