Collared and non-collared manifold boundaries
Mathieu Baillif
math.GN 57N16 (primary)













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lemmaLemma[section]


thm[lemma]Theorem
*thm*Theorem
*lemma?Lemma ??


defi[lemma]Definition
conj[lemma]Conjecture
cor[lemma]Corollary
prop[lemma]Proposition
prob[lemma]Problem
qu[lemma]Question
q[lemma]Questions
*remRemark
examples[lemma]Examples
example[lemma]Example





Collared and non-collared manifold boundaries
    Mathieu Baillif[
Haute école du paysage, d'ingénierie et d'architecture (HEPIA),
département d'Informatique et systèmes de communications,
Genève – Suisse. mathieu.baillif@hesge.ch
]
    
==========================================================================================================================================================================================



We gather in this note results and examples about collared and non-collared boundaries in non-metrisable manifolds.
It is known for a long time that boundaries are collared in metrisable manifolds, but the matter
is more complicated for non-metrisable ones.
The majority of the contents of this note is well known but a bit scattered in the literature (sometimes with small inaccuracies
that we correct), and some of it is apparently not published at all
and might be new.

.2cm
Keywords:  Non-metrisable manifolds, Manifolds boundaries.
.2cm
2010 MSC:  57N16



§ INTRODUCTION AND DEFINITIONS


It is known since the works of M. Brown <cit.> in the sixties
that the boundary ∂ M of a topological metrisable manifold is collared in M, that is, there is a neighborhood U of ∂ M
in M and an embedding h:∂ M×[0,1)→ M sending ⟨ x,0⟩ to x which is a homeomorphism on U. 
(We use brackets ⟨ , ⟩ for ordered pairs, reserving parentheses for open intervals in
ordered sets.)
Brown's result is actually more general
and shows that a locally collared closed subspace of a metrisable space is collared.
Some years later, R. Connelly  <cit.> found another proof that the boundaries of metrisable manifolds are collared
with a very nice argument that works backwards in the sense that you glue the collar to the manifold and then push the manifold little by little inside the collar until
it fills it completely. The collaring embedding is then given by the inverse of this pushing.
Connelly's definition of a collar is slightly more restrictive as the embeddings are assumed to be closed (and have domain ∂ M×[0,1]),
and he gives details only in the case where ∂ M is compact, writing that the proof should work in the case ∂ M is strongly paracompact (see below for a definition).
Since metrisability of M is a priori not necessary for Connelly's argument, with appropriate changes the proof can be adapted to the case where
M is non-metrisable. D. Gauld did exactly this in his recent nice book
<cit.> and showed in particular that if ∂ M is connected and metrisable, then it is collared in M, even 
if M happens to be non-metrisable. 
But looking at the details, we noticed that it is not possible to guarantee that the collar
embeddings are closed (there are counter-examples),
hence the proof shows that the boundary is collared under Brown's definition but not under Connelly's. 



Despite the availability of very good texts about the general theory of non-metrisable manifolds 
(aformentioned Gauld's book <cit.> and the older but less elementary article by P. Nyikos in the Handbook of Set-theoretical Topology 
<cit.>, for instance), we are not aware of a reference gathering all the results and counter-examples pertaining to the collaring problem of the boundary,
hence this small note. 
It can be thought of as a convenient reference for (the few)
researchers investigating nonmetrisable manifolds and/or, for researchers in more usual areas of mathematics, as 
a catalog of the horrors awaiting those who dare to not include metrisability in the definition of 
a manifold[The author 
is a member of the first group and sees these pathologies in a similar way a fungi specialist sees the layers of mold 
which developed in a pasta dish long forgotten in the back of the fridge, but understands the urge of some mathematicians 
(the majority, actually) to flush down the toilet
the entire specimen by imposing metrisability as an unmovable feature of manifolds (and by scheduling a periodic revision of the contents of the fridge).].
Except for small corrections to known results and some of the examples, 
the author claims no originality in the contents of this note.
As such, although we aim to remain readable even for those not used to non-metrisable manifolds,
we do not flesh out completely the details of every argument and construction since they are already available elsewhere, but 
try to convey the main ideas. 
Some familiarity with countable ordinals might be useful in Section <ref>.
We also tried to keep the references to a minimum,
using Gauld's and Nyikos' texts as much as possible, as well as Engelking's book <cit.> for general results in point set topology.



So, we are concerned with boundaries in manifolds, which are assumed to be Hausdorff spaces each of whose points has a neighborhood
homeomorphic to 
^n-1×_≥ 0 for some (fixed) n≥ 1. 
If we want to emphasize the dimension, we say n-manifold or manifold of dimension n.
A surface is a 2-manifold.
The points in the boundary ∂ M are those 
which do not have a neighborhood homeomorphic to ^n, and ∂ M is itself a n-1-manifold without boundary and a closed subset of M. 
We sometimes call M-∂ M the interior of M. The interior in the topological sense will be called “topological interior” to avoid ambiguity.
Manifolds share many properties with the Euclidean space, in particular they are Tychonoff, locally compact and locally connected.
Speaking of connectedness, it is usually included in the definition of manifolds when metrisability is not assumed, mainly because 
for connected manifolds a lot of properties are equivalent to metrisability (see for instance Theorem <ref> below)
and the statements are more cumbersome in the general case.
But we do not assume connectedness in this note, as ∂ M is in general not connected, and we want it to remain a manifold.

The following definition avoids the ambiguity between Connelly's and Brown's by introducing strong collars.
As usual, U denotes the topological closure of U. 


   Let B be a closed subset of a space X. We say that
   B is locally collared in X if there is a cover 𝒰 of B by 
   subsets of B that are open in B and for each U∈𝒰
   an embedding h:U×[0,1]→ X which sends ⟨ x,0⟩ to x when x∈U, such that 
   h^-1(B)=U×{0} and the image of U×[0,1) is open in X. 
   Such an h is called a local collar (of B in X). If the embeddings are closed,
   we say that B is locally strongly collared in X, each h being a local strong collar.
   Finally, if 𝒰={B}, we say that B is collared in X 
   (strongly, if the embedding is closed).

If the ambient space X is clear, we say that B is collared, omitting “in X”.
Context allowing, we sometimes also call “collar” the image of h inside X.
The following theorem summarises the positive results on the collaring problem.
As said above, it is essentially due to M. Brown and R. Connelly.



   Let M be a connected manifold with boundary ∂ M. Then the following hold.

   (a) If ∂ M is compact, then it is strongly collared in M.

   (b) If ∂ M is Lindelöf, then it is collared in M.

   (c) If M-∂ M is metrisable and ∂ M is collared, then ∂ M is Lindelöf.

   (d) If M is normal and ∂ M is countably metacompact (in particular, metrisable or Lindelöf) and collared in M, then ∂ M is strongly collared in M.

   (e) If M-∂ M is metrisable, it has a collared boundary iff it has a strongly collared boundary iff M is metrisable.



Our convention is that normal and regular spaces are Hausdorff.
The definition of a countably metacompact space is recalled in Definition <ref> below.
Recall that a 0-set is the inverse image of {0} for a continuous real-valued function.
In particular, 0-sets are G_δ sets.
The connection with collars is given by the following (easy) theorem.


   Let M be a manifold with boundary ∂ M.

   (a) If ∂ M is strongly collared in M, it is a 0-set in M.

   (b) If M-∂ M is metrisable, then ∂ M is a 0-set in M.

   (c) If ∂ M is collared in M, it is a G_δ set.


Section <ref> is dedicated to the proof of these theorems (and more general results from which they follow). 
Section <ref> contains a catalog of counter-examples (elementary, in majority) to various a priori possible generalisations.
For instance:

– Theorem <ref> (b) may not hold if ∂ M is not Lindelöf, even if M is normal 
(Example <ref>), even if M-∂ M is metrisable and ∂ M connected (Example <ref>), 
even if both M-∂ M and 
∂ M are metrisable (Example <ref>).

– Examples <ref>–<ref> show that even if ∂ M is Lindelöf and thus collared,
   it might be not strongly collared. Hence Theorem <ref> (d) does not hold if M is not normal.

– The boundary is not a G_δ set in Example <ref> (which is normal) and 
   not a 0-set in Examples <ref>–<ref> (which have metrisable boundary and separable interior).
   This bad behaviour can be pushed quite far by having
   closed discrete subsets of the boundary which are not G_δ (Example <ref>). 

– Also, a collared 0-set boundary is not always strongly collared (Example <ref>). 

.3cm
We end this introduction by citing a nice 
result of D. Gauld and S. Greenwood <cit.> (see also <cit.>).
We will not repeat the proof here, but do stress that it relies mainly on two facts:
compact components of the boundary are strongly collared (individually),
and  a metrisable manifold is a countable increasing union of connected open sets with compact closure (see Theorem <ref>).
It is thus an illustration of the use of the collaring theorem for non-metrisable manifolds.
Example <ref> shows that it is not enough to assume that M-∂ M is separable, that is, has a countable dense subset.


  A connected manifold M with metrisable interior M-∂ M
  has at most countably many compact components in its boundary.





This note was written after trying to answer a question on MathOverflow by Kalle Rutanen. We thank him
for the question and his remarks on a draft version of these notes, Martin W. Licht for having spotted 
a small gap in one of our proofs,
as well as David Gauld for useful email exchanges about the subject (and his kindness in general).
We also thank the referee for his/her numerous usefull remarks.









§ THEOREMS



To avoid ambiguity, we reserve the word `boundary' only for the boundary of a manifold and use `frontier' for U-U (when U is open).
We follow the set-theoretic convention of writing ω for the set of natural numbers instead of .
The following long definition contains almost every concept we need in this section.


   If B is a subset of the space X, a B-cover is a family of open subsets of X whose union contains B.
   If ℱ is a B-cover, 
   a B-refinement is a B-cover 𝒢 such that any member of 𝒢 
   is contained in a member of ℱ. If B=X we just say cover and refinement, without “X-”.

   A subspace B of X is metacompact [resp. paracompact] (resp. strongly paracompact) in X iff for any B-cover ℱ
   there is a B-refinement 𝒢 which is point-finite  [resp. locally finite] (resp. such 
   that any member of 𝒢 intersects only finitely many other members) as a family of subsets of X.
   A space is metacompact [paracompact] (strongly paracompact) if metacompact [paracompact] (strongly paracompact) in itself.

   A space is countably metacompact [resp. countably paracompact] 
   if any countable cover has a point-finite [resp. locally finite] refinement.

   A partition of unity is a family of functions λ_α:X→[0,1], α in some index set κ 
   which may be assumed to be a cardinal,
   such that for any x∈ X we have λ_α(x) = 0 for all but finitely many α,
   and ∑_α∈κλ_α(x) = 1.
   The partition of unity is subordinate to a cover of X if the support (that is, the set of points with value ≠ 0)
   of each λ_α is included in a member of the cover.

Notice that if B is closed and metacompact in some space X, then B is metacompact, and ditto for the paracompact and strongly paracompact properties. The converse does not hold as a space can be metacompact (in itself) but not metacompact in
some other space.
We can always assume that a partition of unity 𝒫 subordinate to 
a cover {U_α : α∈κ} 
is given by {λ_α : α∈κ}, where the support of λ_α lies in U_α.
This can be arranged by first taking a choice function c sending each λ∈𝒫
to some α such that U_α contains its support, and then summing those members of 𝒫
with the same image under c. 
Finally, if α∈κ not in the range of c, define λ_α to be constant on 0.
We shall always use such an indexing.
We will need the following classical facts due to Dieudonné and Dowker.



    

   (a) If X is a paracompact Hausdorff 
       space and 𝒰 is a cover of X, then X is a normal 
       space and there is a partition of unity subordinate to 𝒰.
       

   (b) If  𝒰 is a locally finite cover of a normal space X, then there is a partition of unity 
       subordinate to 𝒰.

   (c) A normal space is countably paracompact iff it is countably metacompact 
       iff its product with [0,1] is normal iff its product with  is normal.

   (d) If {U_α : α∈κ} is a cover of a paracompact space X, there is 
       a locally finite refinement {V_α : α∈κ} 
       such that V_α⊂ U_α for each α∈κ.



Let us first investigate in which cases it is possible to obtain (local) strong collars from (local) collars.
First, we note that going from local collars to local strong collars is automatic in regular spaces.
This is not true for Hausdorff spaces in general (see Example <ref>).

   If B is closed and locally collared in the regular space X, then B is locally strongly collared in X.

Given x∈ B and a local collar h:U×[0,1)→ X with image the open set W∋ x, take an open 
   V such that x∈ V⊂V⊂ W.
   Taking a closed strip containing ⟨ x,0⟩ inside h^-1(V), its image is a local strong collar.



When the collar is global, the existence of a strong collar does not automatically follow from the regularity of the space, 
something more is needed.


   Let B be closed and collared in the Hausdorff space M. Assume that B is normal (in the subspace topology) and
   countably metacompact (in itself). Then the following hold. 
   

   (a) If U⊂ M is an open neighborhood of B, 
       there is a collaring embedding g:B×[0,1]→ X whose image is contained in U.

   (b) If M is normal, then B is strongly collared.
   

B is a normal countably metacompact space and is thus countably paracompact by <ref> (c).
   Fix a collaring homeomorphism h:B×[0,1)→ W, where W is an open neighborhood of B.
   

   (a) 
   For each x∈ B, fix q_x∈(0,1)∩ and some open V_x⊂ B
   such that h(V_x×[0,q_x])⊂ U. Consider the countable cover 
   𝒢={G_q  : q∈}, where G_q = ∪_q_x =q V_x,
   and let {λ_q : q∈} be a partition of unity subordinate to it (its existence is secured by Theorem <ref> (b)). 
   We use the partition of unity to glue together the constant functions x↦ q/2 defined in G_q∈𝒢
   to obtain a continuous map, that is, we set f(x)=∑_q∈λ_q(x) · q/2.
   Given 
   x∈ B, 
   λ_q(x)>0 for finitely many q, hence r = max{q∈ : λ_q(x)>0}
   is well defined and
   0<f(x)≤ r. It follows that
   the image by h of C={⟨ x,t⟩∈ B×[0,1] :  t≤ f(x)} is contained in U and
   ⟨ x,t⟩↦⟨ x,t/f(x)⟩ is a homeomorphism between C and
   B×[0,1]. The composition of its inverse with h defines the collaring embedding g.
   (Notice that the image of g is closed in W, but might not be closed in M.)
   
 
   (b) By normality of M let U be open such that B⊂ U⊂U⊂ W.
   By (a), there is a collaring embedding whose image is a closed subset of U⊂ W,
   and hence a closed subset of M.



Let us now turn to our main theorems.
Half of Theorem <ref> is a direct consequence of the following more general result.

   
   Let M be a Hausdorff space and B be a closed subset of M which is locally collared in M and strongly paracompact in M.
   Then B is collared.

Our proof is almost the same as Connelly's in <cit.> and Gauld's version in <cit.>, 
except for some details. 
Some of them are just cosmetic and due to personal taste: we use partitions of unity, reverse some signs and try to avoid formulas,
replacing them by a geometric description.
But two details are actually somewhat important. First, we do not claim that B is strongly collared, even if the local collars are strong, as
it might not be the case in general (see Examples <ref>–<ref>). 
Second, we ask B to be strongly paracompact in M and not just in itself, as the latter
is not sufficient (see Example <ref>). 
Connelly actually only gives a complete proof when B is compact and the local collars are strong
(in which case everything works fine)
and says without giving details that the proof should work in the more general case. Gauld's proof does fill the details, but 
his statement is a little bit imprecise.
Since B is locally 
   collared, we have 
   embeddings h_α:U_α×(-1,0]→ M with h_α(x,0) = x, where {U_α : α∈κ}
   is an open cover of B and κ a cardinal. Let W_α = h_α(U_α×(-1,0]).
   We first show that we can assume that
   each
   W_α meets only finitely many 
   other 
   W_β. 

   By strong paracompactness of B in M we may take a B-refinement 𝒲'={W'_γ : γ∈κ'}
   of 𝒲={W_α : α∈κ} whose members intersect at most finitely other members.
   Notice that W'_γ is a priori not the image of one of our original collaring embeddings, but we 
   now show that we can “shrink” our embeddings in order to have images inside these W'_γ. 
   Set 𝒰'= {U'_γ : γ∈κ'}, where U'_γ= W'_γ∩ B. By Theorem
   <ref> (d), we can shrink a bit each U'_γ to obtain a refinement by open sets V_γ
   such that V_γ⊂ U'_γ.
   Notice that 
   V_γ is a paracompact (hence normal) space lying inside some U_α
   and is thus collared in M.
   Theorem <ref> (a) 
   provides a collaring embedding h'_γ:V_γ×[0,1]→ M whose image lies inside W'_γ.
   Since W'_γ intersects at most finitely many W'_δ,
   the image of h'_γ intersect the image of h'_δ for at most finitely many δ.
   

   We may now forget about these prime symbols and 
   assume that h_α, W_α and U_α are defined as at the beginning of the proof
   and that each
   W_α meets only finitely many 
   other 
   W_β.  
   We let E=∪_α∈κW_α.
   Since B is closed and strongly paracompact in M, it is in particular a paracompact space, 
   hence there is a partition of unity {λ_α : α∈κ}
   subordinate to the cover by the U_α's by Theorem <ref> (a).
   As said above we assume that the support of λ_α lies inside U_α.
   Let f_α:B→[0,1] be given by the partial sum ∑_β≤αλ_β. Then f_α is continuous
   by local finiteness 
   and f_κ(x) = 1 for all x∈ B.

   We glue the space B×[0,1] to M, thus defining the space M^+, 
   by identifying x∈ B⊂ M to ⟨ x,0⟩∈ B×[0,1], and define the local embeddings 
   h_α:B×(-1,1]→ M^+ which extend h_α 
   by the identity in B×[0,1]. Call the glued part the ribbon.
   We now proceed by induction on α to define embeddings Φ_α:E→ M^+ such that
   the image of Φ_α in the ribbon is {⟨ x,t⟩ : t≤ f_α(x)}. 
   (Warning: it might be impossible to obtain an embedding from all of M into M^+, 
   see Examples <ref>–<ref> below.)
   The induction works as follows. 
   We describe geometrically what Φ_α does and refer the reader more confortable with formulas 
   to the proof of <cit.> or to Connelly's original article <cit.> for more details.
   We may first assume f_-1 to be 0, U_-1 to be empty and Φ_-1 to be the identity on E.
   Assume Φ_β is defined for each β<α.
   Since W_α meets only finitely many other local collars 
   we may set β<α to be maximal such that W_α∩ W_β≠∅
   or -1 if W_α intersects no W_β for β<α. 
   By induction the image of Φ_β in the ribbon is given by the points under the graph of f_β.
   Pulling back into U_α×[-1,1] with h_α^-1, we
   are in the situation depicted on the lefthandside of Figure <ref>, where the image of 
   Φ_β inside the collar W_α and the ribbon is in dark grey and the rest of 
   the collar corresponding to W_α is depicted in lighter grey.
   
   To get to the righthandside of the picture, which is what we wish, it is enough to “push along the fibers” in
   U_α×(-1,1] using affine maps sending (-1,f_β(x)] to (-1,f_α(x)] for each x in U_α
   (and using h_α,h_α^-1 to go back and forth).
   Since β<α and we defined f_α and f_β with a partition of unity, f_α and f_β agree
   on the frontier of U_α, so this push moves only the points inside of W_α and defines 
   a continuous embedding of E into M^+.
   

   Since each W_α meets only finitely many other W_γ, we can proceed with the induction until α = κ,
   and local finiteness ensures the continuity of Φ_α at each step.
   In the end each point of E is moved only finitely many times, so letting Φ(y) = Φ_α(y)
   for α maximal such that y∈ W_α, we obtain an embedding
   whose image contains the entire ribbon.
   The preimage of the ribbon minus B is thus a homeomorphic copy of B×[0,1) in E.





Recall that a space is ccc if any family of pairwise disjoint open subsets is at most countable.
A separable space is of course ccc.

   Let X be a Hausdorff space and B⊂ X be closed.
   If X-B is ccc and B is not ccc, then B is not collared in X. 

An uncountable family of disjoint open sets in B yields an uncountable disjoint family of open sets in the collar.



   Let X be a Hausdorff ccc space and B⊂ X be closed and locally connected (in the subspace topology). 
   If B contains uncountably many connected components, then B is not collared.

Any component of B is open and closed in B by local connectedness (see <cit.> if in need of a proof), hence 
   B is a discrete union of its components. 



   Let M be a connected manifold such that M-∂ M is metrisable and ∂ M contains uncountably many components.
   Then ∂ M is not collared.

A connected metrisable manifold is ccc, see just below.



The stage is almost set for the proofs of Theorems <ref>–<ref>, the only missing piece of furniture is a reminder
of the following classical properties of manifolds.

   
   Let M be a connected manifold. Then the following conditions are equivalent and each imply that M is separable and thus ccc. 

   (a) M is metrisable.

   (b) M is Lindelöf.

   (c) M is hereditarily Lindelöf.

   (d) M is paracompact.

   (e) M is strongly paracompact.

   (f) M = ∪_n∈ωK_n where K_n is compact for each n and the topological interior of K_n+1 contains K_n.





[Proof of Theorem <ref>]
    

   (a) The proof of (b) below shows that ∂ M is collared. Since ∂ M×[0,1/2] is compact, its homeomorphic image in M is closed.

   (b) 
   Since ∂ M is Lindelöf, we may cover it by (at most) countably many
   open sets homeomorphic to ^n-1×_≥ 0. The union of these open sets gives a submanifold N⊂ M with ∂ N =∂ M. Since 
   N is Lindelöf it is metrisable and thus strongly paracompact by Theorem <ref>. 
   Of course this implies that ∂ M is strongly paracompact in N, and we conclude with Theorem <ref>.
   

   (c) Since ∂ M is collared, there is a homeomorphic copy of ∂ M inside M-∂ M. 
       By Theorem <ref>, M-∂ M is hereditarily Lindelöf, and thus so is ∂ M.

   (d) This is Theorem <ref> (b). 
       (Recall that a Lindelöf Tychonoff space is paracompact, see e.g. <cit.>.)

   (e) 
       Theorem <ref> tells us that Lindelöfness is equivalent to metrisability for connected manifolds. 
       Assume M-∂ M to be metrisable. 
       Then a collared boundary implies by (c) that M is metrisable, and hence normal and hereditarily paracompact,
       so the boundary is strongly collared by (d).





[Proof of Theorem <ref>]
      

   (a) The collar is homeomorphic to B×[0,1] and closed in M, define 
   f to be the second coordinate `in the collar' and 1 elsewhere. 
   This defines a continuous function, because since ∂ M is a manifold without boundary,
   any point in the frontier of the collar is on its “upper part” in the interior of M.

   (b)
   Let M be a manifold with metrisable interior. By Theorem <ref> (f), M-∂ M is
   a union ∪_n∈ωK_n where each K_n is compact and the topological interior of K_n+1
   contains K_n for each n.
   Hence (using either the metric or Urysohn's lemma) there is a continuous
   f:M-∂ M→_+  such that f(K_n+1-K_n)⊂[1/n+2,1/n+1]. 
   Set f:M→_+ to be 0 on ∂ M and equal to f outside of it.
   Then f is continuous and ∂ M is the preimage of {0}.

   (c) Let h:∂ M×[0,1)→ M be a collaring embedding.
       Then ∂ M, as a subset of M, is the countable intersection of the open sets h( ∂ M×[0,1/n)).









§ EXAMPLES (GALLERY OF HORRORS)



Almost all our examples are classical, although they were generally introduced in contexts 
totally unrelated to the collaring problem. We tried to give the name (if known to us)
of who first came up with the example in question (if non trivial).



[Trivial]
   A non-metrisable manifold with strongly collared boundary.

[Details]
   Choose your favorite non-metrisable manifold M without boundary (for example _+ defined below). Then
   M× [0,1) trivially has a strongly collared boundary. 


As usual, ω_1 denotes the first uncountable ordinal.
Any ordinal is given by the set of its predecessors, so in particular ω_1 is the set of all countable ordinals.
A countable subset of ω_1 has a supremum which is a countable ordinal,
this elementary fact will be used (implicitly) many times in this section.
Recall that the longray _≥ 0 is the 1-manifold ω_1×[0,1) endowed
with the topology given by the lexicograpic order. Its boundary contains only the point ⟨ 0,0⟩,
removing it we obtain the open longray _+. Chapter 1.2 in <cit.> is dedicated to
proving almost all the elementary properties of _+ and _≥ 0, 
recall in particular that both are normal and _≥ 0 is countably compact.
We sometimes consider ω_1 as a subspace of _≥ 0 by identifying α∈ω_1 with ⟨α,0⟩∈_≥ 0.
The following lemma is well known.
Recall that a subset of ω_1 (or _+) is stationary if it meets any closed and unbounded (abbreviated club) 
subset of ω_1  (or _+).
A club subset of ω_1 (or _+) is stationary (see e.g. <cit.>)
and contains a copy of ω_1.
The definitions of stationary and club extend in an obvious way
to such sets as horizontal or vertical lines and the diagonal of _+^2.
As one may guess, [α,ω_1) is the set of points of _+ or _≥ 0 that are ≥α,
and other intervals are defined similarly.
For any x<y in _+, the interval (x,y) is homeomorphic to .


      

   (a) If U is an open subset of _+^2 whose intersection with the diagonal is stationary, 
       then
       U contains [α,ω_1)^2 for some α∈ω_1.

   (b) If U is an open subset of _+× [0,1] whose intersection with the horizontal line _+×{a}
       is stationary, then U contains [α,ω_1)×(a-1/n,a+1/n) 
       for some α∈ω_1,n∈ω.
       

   (c) Let C⊂_+^2 be closed.
       If C∩ [α,ω_1)^2≠∅ for each α∈ω_1, 
       then C intersects the diagonal in a club
       set. If C∩ [α,ω_1)^2=∅ for some α, 
       then C intersects either some horizontal line _+×{a} or some vertical line {a}×_+
       in a club
       set. 

   (d) _+×, _≥ 0×, _+^2 and (_≥ 0)^2 are normal spaces.

The elementary proofs of (a) to (c) are essentially done in <cit.>. 
   We now discuss the normality claims.
   It is easy to check that since _+ = (0,1)∪[1,ω_1) is the union of a Lindelöf and a 
   countably compact space, it is countably paracompact.
   By Theorem  <ref> (c) _+× is normal.
   Now, assume that A,B are disjoint closed sets of _+^2.
   Recall that the intersection of two (actually, countably many) club subsets of _+ is again club   
   <cit.>.
   The diagonal is a copy of _+, by (c) one of A,B, say B,
   does not intersect [α,ω_1)^2 for some α.
   Hence B⊂ (0,α)×_+ ∪_+×(0,α), and
   we may finish by using the result for _+×.
   The proofs for _≥ 0 are almost the same.


[Folklore]    
    
    A normal surface with a boundary which is not a G_δ (and hence is non-collared).

[Details]
  The closed octant 𝕆={⟨ x,y⟩∈(_≥ 0)^2 : y≤ x} has a boundary 
  which is not a G_δ. Indeed, 
  by Lemma <ref> (a) any open set containing
  the diagonal contains {⟨ x,y⟩ : α≤ y≤ x} for some α,
  and this remains true if one takes a countable intersection.
  As a closed subspace of a normal space, 𝕆 is normal.


[Folklore]
    A surface with a collared 0-set boundary which is not strongly collared.
      

[Details]
   Set S = _+×[0,2) - ω_1×{0} (where ω_1 is seen as a subset of _≥ 0). 
   The boundary (_+-ω_1)×{0}
   is an uncountable discrete union of open intervals, is collared (with collar (_+-ω_1)×[0,2)) and is a 0-set,
   but is not strongly collared. 
   To see this, let h:(_+-ω_1)×[0,1]→ S be a strong collar and W be its closed image in S.
   Any function ω_1→^2 is eventually constant, meaning that it is constant in [α,ω_1) for some α,
   and a countably compact subset of a manifold is closed (see e.g. <cit.>).
   It is easily seen to imply that
   (_+-ω_1)×[0,2) does not contain a copy of ω_1. Hence,
   W is bounded in each horizontal line _+×{a}, a∈(0,2) and 
   thus the open set S-W contains [β_a,ω_1)×(a-1/n_a,a+1/n_a) 
   for some β_a,n_a by Lemma <ref> (b).
   By Lindelöfness, let E⊂ be countable such that ∪_a∈ E (a-1/n_a,a+1/n_a) covers (0,2) and
   set β=sup_a∈ Eβ_a. Then S-W⊃ [β,ω_1)× (0,2).
   This is clearly impossible as boundary components beyond β would not be collared.
Our next examples are based on procedures to add (or delete) boundary components in surfaces called Prüferisation, Moorisation and Nyikosisation.
They are described in great detail in <cit.>, we will thus only give a geometric idea of the constructions.


Start with a half plane _≥ 0×, and choose a point ⟨ 0,a⟩ on the vertical axis.
We replace this point by a new boundary component _a, which is a copy of , as follows.
A basic
neighborhood of some point x∈_a is given by the union of an interval (u,v)⊂_a containing x and a triangle 
comprised between the lines of slopes u and v pointing at 
⟨ 0,a⟩ and the vertical line {w}×, for some w>0. 
We then say that we have Prüferised the half plane at height (or at point) a. 
Figure <ref> is a pictural description. 
The new boundary then consists of the old boundary minus {⟨ 0,a⟩} together with
the new boundary component _a. It takes a moment's thought to accept that we can 
perform this Prüferisation at each point in the vertical axis and still have a surface
(which is the original Prüfer surface, actually), because
the added boundary components are independant of each other, so to say. We can also Prüferise only at some (but not all) points
and still obtain a surface, as long as 
we remove the rest of the initial boundary if 
the set of a where the Prüfersiation
is done is not closed.
Moreover, the added boundary components form a discrete family: 
The union of _a and an open disk tangent to the vertical line at ⟨ 0,a⟩ is a neighborhood
of _a (and intersects only this boundary component).


The process of Moorisation at point a consists of first Prüferising at a and then identify x with -x in the added boundary component _a.
The points in _a are in a sense pushed inside the interior of the surface.
As before, we can also Moorise at each point at once, or just some of them. 

The Nyikosisation at point a is done similarly, but this time we add a boundary component which is a copy of _+.
Instead of taking triangles, we take a family of curves c_α (α∈ω_1) pointing at ⟨ 0,a⟩
such that if β<α, then close enough to the vertical axis c_α is above c_β. 
(Such a family of curves can be easily constructed from a family of functions f_α:ω→ω such that 
f_β<^* f_α when β<α, where <^* means `smaller outside of a finite set'. 
See <cit.> for more details.)
Then a neighborhood of a point in the added longray is given by an interval in the longray union the space between 
two curves (corresponding to the endpoints of the interval) and a vertical line. 
In particular, c_α has α∈_+ as its unique limit point. 

Notice that the resulting surface in any of these constructions is separable, as the interior of the half plane is dense.





[Rado]
   A separable surface with metrisable interior and a non-collared metrisable boundary which is strongly paracompact in itself.
   

[Details]
   Prüferise the half plane at each height. The resulting surface has then 
   uncountably many boundary components and thus a non-collared boundary by Corollary <ref>.
   The boundary is made of a discrete uncountable collection of copies of  and is thus metrisable
   and strongly paracompact in itself.


[Nyikos]
    A separable surface with a non-collared single boundary component and metrisable interior.
      

[Details]
   The Nyikosisation (at any point)
   of the half plane has a non-collared boundary by Theorem <ref> (c).


The boundary is a 0-set in the two previous examples by Theorem <ref> (b).
We shall now give an example
of a manifold where it is not the case. First we need
the following fact. We were not able to spot a reference for a proof, so we give one.


   Let ℙ be the surface obtained by Prüferising the positive half plane at each height
   and let ∂ℙ=∪_a∈_a.
   Let G⊂ be a non-meagre set (hence, in particular, somewhere dense). Let U be an open subset of ℙ such that
   for each a∈ G, there is some x_a∈_a∩ U. 
   Then U contains a `strip' along the boundary (0,c)×(u,v) (where c,u,v∈, c>0, u<v).

   

   .65
   Let {q_n : n∈ω} 
   be an enumeration of . Given m,n,k∈ω such that q_k>q_n and a∈,
   let T_k,n,m,a be the interior of the triangle in the positive half plane shown on the right.
   Given a∈ G, there is a point x_a in U∩_a and thus m,n,k∈ω
   such that T_k,n,m,a⊂ U. Let 
   
    G_k,n,m = {a ∈ G :  T_k,n,m,a⊂ U}.

   Since G is the union of the G_k,n,m, one of them must be non-meagre,
   so there is an interval (u,v)⊂ in which G_k,n,m is dense.
   Since the slopes q_n,q_k and m are fixed, U contains a parallelogram of width 1/m.
   The rest follows.
   
     .3
     figure = triangle.eps, width = .6
   


Notice that this lemma also holds if some of the boundary components are Moorised.

   A separable surface S with a non 0-set hence non-strongly collared, but metrisable and collared, boundary ∂ S.
   Moreover, ∂ S is strongly paracompact in S.
   

[Details]
   Let S be the surface S defined by Prüferising 
   the half plane at each height and then Moorising the boundary components _a
   for a∈-. The boundary being ∪_a∈_a, it is collared by Lindelöfness and Theorem <ref>.
   The argument given in the proof of Theorem <ref> (b) above shows that Lindelöfness also implies that
   ∂ S is strongly paracompact in S.
   Suppose that f:S→[0,1] witnesses that ∂ S is a 0-set.
   For a∈-, the Moorised boundary components _a of ℙ are sent in (0,1] by f.
   Hence there is some n such that U_n = f^-1((1/n,1]) intersects the Moorised _a for a non-meagre set of a.
   By Lemma <ref> there is some strip (0,c)×(u,v) inside of U_n.
   The image of f on the closure of U_n is contained in [1/n,1]. But
   _a is in this closure for some rational a, which contradicts the fact that it is a boundary component of S
   on which f should be 0.


We note in passing that the (double of the) surface of Example <ref> also appears in Appendix B of
<cit.> in which D. Fernández-Bretón and N.G. Vlamis study ends of non-metrisable manifolds.


   A separable surface with a non 0-set hence non-strongly collared, but metrisable, collared and connected boundary,
   which is strongly paracompact in the whole surface.
   

[Details]
   By identifying some points in the boundary in Example <ref>.
   Let q_n, n∈ω be an enumeration of . 
   If B⊂, we write B_q for the corresponding subset of _q (the boundary component at height q).
   We identify now x∈ [1,∞)_q_n with -x∈(-∞,-1]_q_n+1. 
   The resulting boundary is a copy of . See Figure <ref>.


[R.L. Moore, essentially]
   A separable surface with uncountably many circle boundary components.
   

[Details]
   Take the surface of Example <ref> obtained by Prüferising the half plane at each height.
   In each boundary component _a identify x with -x
   when x≥ 1. (See Figure <ref>.) The non-identified points in each _a together with 1
   (glued to -1) now form
   a circle.
   The rest of _a ends in the interior of the surface, which has thus a non-metrisable (but separable) interior.







Let us give two last examples of manifolds with bad behaviour on the boundary by showing that even closed discrete subsets
can exhibit a reluctance to act decently.



   The boundary of the surface of Example <ref>
   contains a countable closed discrete subset which is not a 0-set.

[Details]
   For each a∈ choose one point in the boundary component _a. This defines a closed discrete set
   which is not a 0-set, the proof being the same as the one given in Example <ref>.


However, any closed discrete subset of this surface is a G_δ. 
(This can be seen roughly by first noting that any uncountable closed discrete subset is
contained in the boundary except for countably many points. Then, 
as noted in 3.5 the boundary is not collared but of course it is a G_δ,
and we can argue more or less inside each boundary component. We skip the details.)

This is not the case in the next example.


   A surface with metrisable non-collared boundary which contains a closed discrete subset which is not a G_δ (hence not a 0-set).

A variant of this space is cited in <cit.>, which is one of the reasons we chose to include it here.
Although we could not spot one in the literature, such examples are known for a long time,
see for instance <cit.>. (A normal example is way more elusive, since none exist
under various set-theoretic assumptions, for instance 𝐕=𝐋.)
The proof of the claimed properties is based on the well known Fodor's Lemma (also known as Pressing Down Lemma), 
which we recall below, and whose proof can be found in any book about set theory, for instance
<cit.>. 


   Let S be a stationary subset of ω_1 and f:S→ω_1 be such that f(α)<α for each α∈ S.
   Then there is α such 
   that f^-1({α}) is stationary.



[Details of Example <ref>]
   We start with the octant 𝕆 of Example <ref>, 
   and Prüferise it on the diagonal as follows. 
   The idea is to add new boundary components (real lines)
   at points ⟨α,α⟩ for α∈ω_1 limit. What happens to the rest of the boundary is irrelevant
   hence we might as well delete it.
   We add these components in such a way that the vertical line {α}×[0,α) 
   converges to the 0 point of the real line attached at ⟨α,α⟩.
   We might do it as follows.
   For each limit α∈ω_1, choose a strictly increasing sequence α_n∈_+ converging to it.
   In what follows, the intervals (0,α),(0,α] are understood as being in _+ if α≥ω, 
   otherwise they are 
   intervals of .
   Fix a homeomorphism Φ:(0,α] → (0,1] sending α_n to 1-1/n, and extend it to a 
   homeomorphism (0,α+1) → (0,2)
   the obvious way.
   This yields a homeomorphism Ψ:(0,α+1)^2→(0,2)^2. 
   In {⟨ x,y⟩∈(0,2)^2 : y< x}, Prüferise at ⟨ 1,1⟩ such that the 0 point 
   of the added real line is the limit of the vertical line below ⟨ 1,1⟩, and pull back this Prüferisation
   in the octant by Ψ^-1. 
   Figure <ref> tries to depict what we mean by showing a 
   neighborhood of 0 pulled back in the octant. This defines a surface, we call it S. 
   We write 0_α for the 0 point in the component added at ⟨α,α⟩. 
   

   
   
   
   Then D={0_α : α∈ω_1,α limit} is a closed discrete subset of S: the 
   intersection of the neighborhood depicted in Figure <ref> with D is {0_α}.
   We now prove that D is not a G_δ by showing that for any open set U containing D, there is β∈ω_1
   such that for each γ≥β, U⊃ [ζ,ω_1)×{γ} for some ζ<ω_1.
   This property still holds if we take a countable intersection, proving the result.
   In passing, it also shows that the boundary is not collared.
   

   Let thus U⊃ D, and for each limit α∈ω_1, choose β(α)<α 
   such that the vertical segment {α}×[β(α),α) lies in U.
   (It exists by definition of neighborhoods of 0_α.)
   By Fodor's lemma <ref>, 
   there is β<ω_1 such that β(α) = β for a stationary set E⊂ω_1 of α.
   This means that if we take γ≥β, U 
   contains ⟨α,γ⟩ for each α>β in E, hence for each such α
   there is ζ(α)<α
   with [ζ(α),α]×{γ}⊂ U. Applying Fodor again, 
   we see that U⊃ [ζ,ω_1)×{γ} for some ζ<ω_1, which is what we wanted to prove.

   Those who enjoy pictures may consult Figure <ref> for a graphic 
   summary of this double application of Fodor's lemma.


Our last example is not a manifold but shows that Lemma <ref> does not hold when the space is not assumed to be regular.




   
   A Hausdorff space X with a closed locally collared subset B which is not locally strongly collared.

[Details]
   Let H=[0,1)^2 be given the half disk topology <cit.>, that is, neighborhoods of 
   points in [0,1)×(0,1) are usual open sets in the half plane
   and neighborhoods of p=⟨ x,0 ⟩ are the union of {p}
   and an open disk centered at p (or any open neighborhood of p in the plane) intersected with [0,1)×(0,1).
   Then H is a Hausdorff non-regular space.
   Let X be H×[0,1) (endowed with product topology) with E=(0,1)×{0}×{0} removed.
   Set B to be the subset of points with 0 third coordinate. 
   Then B is closed and {⟨ x,y,z⟩∈ X : ⟨ x,y,0⟩∈ B} is an open collar.
   Since we removed E, the topology on B is the usual topology as a subset of the plane, 
   in particular the product of any of its subsets with [0,1) is regular.
   (Recall that any product of regular spaces is regular, see e.g. in <cit.>.)
   But any closed neighborhood of x=⟨ 0,0,0⟩∈ B is not regular because, as seen in Figure <ref>,
   it will contain a piece of a line [0,a]×{0}×{c} for some a,c.
   Hence such a closed neighborhood cannot be homeomorphic to 
   U×[0,1] for any U⊂ B.  There is thus 
   no closed local collar 
   at x. 





10

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