aa r X i v : . [ m a t h . GN ] M a r Collared and non-collared manifold boundaries
Mathieu BaillifMarch 27, 2020
Abstract
We gather in this note results and examples about collared or non-collared boundaries of non-metrisable manifolds. Almost everything is well known but a bit scattered in the literature, and some ofit is apparently not published at all.
It is known since the works of M. Brown [1, Theorems 1 & 2] a long time ago 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 , → M sending h x, i to x which is an homeomorphismon U . (We use brackets h , i for ordered pairs, reserving parenthesis for open intervals in orderedsets.) Brown’s result is actually more general and shows that a locally collared closed subspaceof a metrisable space is collared. Some years later, R. Connelly [2] found another proof that theboundaries of metrisable manifolds are collared with a very nice argument that works backwardsin the sense that you glue the collar to the manifold and then push the manifold little by littleinside the collar until it fills it completely. The collaring embedding is then given by the inverseof this pushing. Connely’s definition of a collar is slightly more restrictive as the embeddings areassumed to be closed (and have domain ∂M × [0 , ∂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 Connely’sargument, 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 [4, Theorem 3.10 & Corollary3.11] 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, as there are counter-examples,hence the proof shows that the boundary is collared under Brown’s definition but not underConnelly’s.Despite the availability of very good texts about the general theory of non-metrisable mani-folds (the aformentioned Gauld’s book [4] and the older but less elementary article by P. Nyikosin the Handbook of Set-theoretical Topology [7], for instance), we are not aware of a refer-ence gathering all the results and counter-examples pertaining to the collaring problem of theboundary, hence this small note. It can be thought of as a convenient reference for (the few)researchers using nonmetrisable manifolds and/or, for researchers in more usual areas of math-ematics, as a catalog of the horrors awaiting those who dare to not include metrisability in thedefinition of a manifold . Except for small corrections to known results, the author claims no The author is a member of the first group and sees these pathologies in a similar way a fungi specialist sees thelayers of mold which developped in a pasta dish long forgotten in the back of the fridge, but understands the urge ofsome mathematicians (the majority, actually) to flush down the toilet the entire specimen by imposing metrisabilityas an unmovable feature of manifolds (and by scheduling a periodic revision of the contents of the fridge). riginality in the contents of this note. As such, although we aim to remain readable even forthose not used to non-metrisable manifolds, we do not flesh out completely the details of everyargument and construction since they are already available elsewhere, but try to convey themain ideas. We also tried to keep the references to a minimum, using Gauld’s and Nyikos’ textsas much as possible, as well as Engelking’s book [3] for general results in point set topology.We are thus concerned with boundaries in manifolds, which are assumed to be Hausdorffspaces each of whose points has a neighborhood homeomorphic to R n − × R ≥ . If we wantto emphasize the dimension, we say n -manifold or manifold of dimension n . A surface is a2-manifold. The points in the boundary ∂M are those which do not have a neighborhoodhomeomorphic to R n , and ∂M is itself a n − M . We sometimes call M − ∂M the interior of M . The interior in the topological sensewill be called “topological interior” to avoid ambiguity. Manifolds share many properties withthe Euclidean space, in particular they are Tychonoff, locally compact and locally connected.Speaking of connectedness, it is usual to include it in the definition of manifolds when metris-ability is not assumed, mainly because for connected manifolds a lot of properties are equivalentto metrisability (see for instance Theorem 2.9 below) and the statements are more cumbersomein the general case. But we do not assume connectedness in this note, as ∂M is in general notconnected, and we want it to remain a manifold.The following definition avoids the ambiguity between Connelly’s and Brown’s by introducingstrong collars. As usual, U denotes the closure of U . Definition 1.1.
Let B be a closed subset of a space X . B is locally collared in X if there isan open cover U of B and for each U ∈ U an embedding h : U × [0 , → X which sends h x, i to x when x ∈ U , such that h − ( B ) = U × { } and the image of U × [0 , is open in X . Suchan h is called a local collar (of B in X ). If the embeddings are closed, we say that B is locallystrongly collared in X , each h being a local strong collar. Finally, if U = { B } , we say that B iscollared or strongly collared in X . 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 summarisesall the positive results on the collaring problem for manifold boundaries. As said above, it isessentially due to M. Brown and R. Connelly. Theorem 1.2.
Let M be a connected manifold with boundary ∂M . Then the following holds.(a) If ∂M is compact, then it is strongly collared in M .(b) If ∂M is Lindel¨of, then it is collared in M .(c) If M − ∂M is metrisable and ∂M is collared, then ∂M is Lindel¨of.(d) If M is normal and ∂M is countably metacompact (in particular, metrisable or Lindel¨of )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 boundaryiff M is metrisable. Our convention is that normal and regular spaces are Hausdorff. The definition of a count-ably metacompact space is recalled in Definition 2.1 below. Recall that a 0 -set in a space is theinverse image of { } for a continuous real-valued function. The connection with collars is givenby the following (easy) theorem. Theorem 1.3.
Let M be a manifold with boundary ∂M .(a) If ∂M is strongly collared in M , it is a -set in M .(b) If M − ∂M is metrisable, then ∂M is a -set in M . Section 2 is dedicated to the proof of these theorems (and more general results from whichthey follow). Section 3 contains a catalog of counter-examples (elementary, in majority) tovarious a priori possible generalisations. For instance: Theorem 1.2 (b) may not hold if ∂M is not Lindel¨of, even if M is normal (Example 3.3),even if M − ∂M is metrisable and ∂M connected (Example 3.6), even if both M − ∂M and M are metrisable (Example 3.5).– Examples 3.8–3.9 show that even if ∂M is Lindel¨of and thus collared, it might not bestrongly collared (showing in passing that Theorem 1.2 (d) does not hold if M is not normal).– The boundary is not a 0-set in Example 3.3 (which is normal) and Examples 3.8–3.9 (whichhave metrisable interior) and this bad behaviour can be pushed quite far (Example 3.12).– Also, a collared 0-set boundary is not always strongly collared (Example 3.4).We end this introduction by citing a nice result of D. Gauld. We will not repeat his proofhere, but do stress that it relies mainly on two facts: compact components of the boundary arestrongly collared (individually), and a metrisable manifold is a countable increasing union ofconnected open sets with compact closure (see Theorem 2.9). It is thus an illustration of theuse of the collaring theorem for non-metrisable manifolds. Example 3.10 shows that it is notenough to assume that M − ∂M is separable , that is, has a countable dense subset. Theorem 1.4 (D. Gauld [4, Proposition 3.12]) . A connected manifold M with metrisable inte-rior 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 thanks him for the question and his remarks on a draft version of these notes, as well asDavid Gauld for useful email exchanges about the subject (and his kindness in general).
To avoid ambiguity, we reserve the word ‘boundary’ only for the boundary of a manifold anduse ‘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 N . The following definition contains almost everyconcept we need. Definition 2.1. If B is a subset of the space X , a B -cover is a family of open sets whose unioncontains B . If F is a B -cover, a B -refinement is a B -cover G such that any member of G iscontained 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 ifffor 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 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 apoint-finite [resp. locally finite] refinement.A partition of unity is a family of functions λ α : X → [0 , , α in some index set κ which maybe assumed to be a cardinal, such that for any x ∈ X we have λ α ( x ) = 0 for all but finitely many α , and P α ∈ κ λ 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 (usually indexedwith the same α ) of the cover. Notice that if B is closed and metacompact in some space X , then B is metacompact, andditto for the paracompact and strongly paracompact properties. We will need the followingclassical facts due to Dieudonn´e and Dowker. Theorem 2.2 ([3, Theorems 5.1.9, 5.2.6, 5.2.8 & Exercice 5.2.A], for instance) . (a) If X is a paracompact T space and U is a cover of X , then X is a normal space and thereis 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 subordinateto U .(c) A normal space is countably paracompact iff it is countably metacompact iff its product with [0 , is normal iff its product with R is normal. Most of Theorem 1.2 is a direct consequence of the following more general result.
Theorem 2.3 (R. Connelly) . Let M be a Hausdorff space and B be a closed subset of M whichis locally collared in M and strongly paracompact in M . Then B is collared. Our proof is almost the same as Connelly’s in [2] and Gauld’s version in [4, Theorem 3.10],except for some details. Some of them are just cosmetical and due to personal taste: we usepartitions of unity, reverse some signs and try to avoid formulas, replacing them by a geometricaldescription. 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.Second, we ask B to be strongly paracompact in M and not just in itself. Connelly actuallyonly gives a complete proof when B is compact and the local collars are strong (in which caseeverything works fine) and says without giving details that the proof should work in the moregeneral case. Gauld’s proof does fill the details, but his statement is a little bit unprecise. Proof.
Since B is locally collared, we have embeddings h α : U α × ( − , → M with h α ( x,
0) = x ,where { U α : α ∈ κ } is an open cover of B , κ being a cardinal. Let W α = h α ( U α × ( − , M that each W α meets only finitely many other W β . We let E = ∪ α ∈ κ W α . Since B is closed and strongly paracompact in M , it is in particulara paracompact space, hence there is a partition of unity { λ α : α ∈ κ } subordinate to the coverby the U α ’s by Theorem 2.2 (a). We of course assume that the support of λ α is inside U α . Let f α : B → [0 ,
1] be given by the partial sum P β ≤ α λ β . Then f α is continuous by local finitenessand f κ ( x ) = 1 for all x ∈ B .We glue the space B × [0 ,
1] to M , thus defining the space M + , by identifying h x, i and x ,and define the local embeddings b h α : B × ( − , → M + which extend h α by the identityin B × [0 , the ribbon . We now proceed by induction on α to defineembeddings Φ α : E → M + such that the image of Φ α in the ribbon is {h x, t i : t ≤ f α ( x ) } .( Warning: it might be impossible to obtain an embedding from all of M into M + , see Examples3.8–3.9 below.) The induction works as follows. We describe geometrically what Φ α does andrefer the reader more confortable with formulas to the proof of [4, Theorem 3.10] or to Connelly’soriginal article [2] for more details. We may first assume f to be 0, U to be empty and Φ tobe the identity on E . Assume Φ β is defined for each β < α . Since W α meets only finitely manyother collars we may set β < α to be maximal such that W α ∩ W β = ∅ or 0 if W α intersectsno W β for β < α . By induction the image of Φ β in the ribbon is given by the points under thegraph of f β . Pulling back into U α × [ − ,
1] with b h − α , we are in the situation depicted on thelefthandside of Figure 2, where the image of Φ β inside the collar W α and the ribbon is in darkgrey and the rest of the collar corresponding to W α is depicted in lighter grey. To get to therighthandside of the picture, which is what we wish, it is enough to “push along the fibers” in U α × ( − ,
1] using affine maps sending ( − , f β ( x )] to ( − , f α ( x )] for each x in U α (and using b h α , b h − α to go back and forth). Since β < α and we defined f α , 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 definesa continuous embedding of E into M + .Since each W α meets only finitely many other W γ , the induction can be carried until α = κ , thelocal finiteness ensures the continuity of Φ α at each step. In the end each point of E is movedonly finitely many times, so letting Φ( y ) = Φ α ( y ) for α maximal such that y ∈ W α , we obtainan embedding whose image contains the entire ribbon. The preimage of the ribbon minus B isthus an homeomorphic copy of B × [0 ,
1) in E . We note in passing that going from local collars to local strong collars is automatic in regularspaces. This is not true for Hausdorff spaces in general (see Example 3.14).
Lemma 2.4. If B is closed and locally collared in the regular space X , then B is locally stronglycollared in X .Proof. Given x ∈ B and a local collar h : U × [0 , → X with image the open set W ∋ x , takean open V ∋ x such that V ⊂ W . Take a closed strip containing h x, i inside h − ( V ), its imageis a local strong collar.When the collar is global, the existence of a strong collar does not automatically follow fromthe regularity of the space, but it does follow from its normality. Theorem 2.5.
Let B be closed, countably metacompact (in itself ) and collared in the normalspace M . Then B is strongly collared.Proof. B is a normal countably metacompact space and is thus countably paracompact by 2.2(c). Fix a collaring homeomorphism h : B × [0 , → W , where W is an open neighborhood of B .By normality of M let U be open such that B ⊂ U ⊂ U ⊂ W . For each x ∈ B , fix q x ∈ (0 , ∩ Q and some open V x ⊂ B such that h ( V x × [0 , q x ]) ⊂ U . Take a locally finite refinement O ofthe countable cover G = { G q : q ∈ Q } , where G q = ∪ q x = q V x , and fix a partition of unity O ( O ∈ O ) subordinate to it (its existence is secured by Theorem 2.2 (b)). For each O ∈ O choose G q ∈ U containing it and set a O = q . We use the partition of unity to glue togetherthe constant functions y a O defined in O ∈ O to obtain a continuous map, that is, we set f = P O ∈O λ O a O . Given y ∈ B , y belongs to finitely many members of O . If q is the maximalvalue of a O for those O ’s containing y , then 0 < f ( y ) ≤ q . It follows that the image by h of C = {h x, t i : t ≤ f ( x ) } is contained in U and h x, t i 7→ h x, t/f ( x ) i is a homeomorphism between C and B × [0 , h ( C ) is a closed subset of W contained in U ⊂ W , hence is a closedsubset of M .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. Theorem 2.6.
Let X be a Hausdorff space and B ⊂ X be closed. If X − B is ccc and B is notccc, then B is not collared in X .Proof. An uncountable family of disjoint open sets in B yields an uncountable disjoint familyof open sets in the collar. Corollary 2.7.
Let X be a Hausdorff ccc space and B ⊂ X be closed and locally connected (inthe subspace topology). If B contains uncountably many connected components, then B is notcollared.Proof. Any component of B is open and closed in B by local connectedness (see [3, Exercise6.3.3] if in need of a proof), hence B is a discrete union of its components. Corollary 2.8.
Let M be a connected manifold such that M − ∂M is metrisable and ∂M contains uncountably many components. Then ∂M is not collared.Proof. A metrisable manifold is ccc, see just below.The stage is almost set for the proofs of Theorems 1.2–1.3, the only missing piece of furnitureis a reminder of the following classical properties of manifolds.
Theorem 2.9 ([4, Theorems 1.2 and A.15]) . Let M be a connected manifold. Then the follow-ing conditions are equivalent and each imply that M is separable and thus ccc.(a) M is metrisable.(b) M is Lindel¨of.(c) M is hereditarily Lindel¨of.(d) M is paracompact.(e) M is strongly paracompact.(f ) M = ∪ n ∈ ω K n where each n is compact and the topological interior of K n +1 contains K n .Proof of Theorem 1.2. (a) The proof of (b) below shows that ∂M is collared. Since ∂M × [0 , ] is compact, itshomeomorphic image in M is closed.(b) Since ∂M is Lindel¨of, we may cover it by (at most) countably many open sets homeomorphicto R n − × R ≥ . The union of these open sets gives a submanifold N ⊂ M with ∂N = ∂M .Since N is Lindel¨of it is metrisable and thus strongly paracompact by Theorem 2.9. Of coursethis implies that ∂M is strongly paracompact in M , and we conclude with Theorem 2.3.(c) Since ∂M is collared, then there is a homeomorphic copy of ∂M inside M − ∂M . By Theorem2.9, M − ∂M is hereditarily Lindel¨of, and thus so is ∂M .(d) This is Theorem 2.5. (Recall that a Lindel¨of Tychonoff space is paracompact, see e.g. [3,Theorem 5.1.24].)(e) Lindel¨ofness is equivalent to metrisability for connected manifolds. Assume M − ∂M to be etrisable. Then a collared boundary implies by (c) that M is metrisable, and hence normaland hereditarily paracompact, so the boundary is strongly collared by (d). Proof of Theorem 1.3. (a) The collar is homeomorphic to B × [0 ,
1] and closed in M , define f to be the secondcoordinate ‘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.(b) Let M be a manifold with metrisable interior. By Theorem 2.9 (f), M is a union ∪ n ∈ ω K 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 → R + suchthat f ( K n +1 − K n ) ⊂ [ n +2 , n +1 ]. Set b f : M → R + to be 0 on ∂M and equal to f outside of it.Then b f is continuous and ∂M is the preimage of { } . Almost all our examples are classical, although they were generally introduced in contexts totallyunrelated to the collaring problem. We tried to give the name (if known to us) of who firstcame up with the example in question (if non trivial).
Example 3.1 (Trivial) . A non-metrisable manifold with strongly collared boundary.Details.
Choose your favorite non-metrisable manifold M without boundary. Then M × [0 , ω denotes the first uncountable ordinal. Recall that the longray L ≥ is the 1-manifold ω × [0 ,
1) endowed with the topology given by the lexicograpic order. Its boundarycontains only the point h , i , removing it we obtain the open longray L + . Chapter 1.2 in [4] isdedicated to proving almost all the elementary properties of L + , recall in particular that it is anormal countably compact space. We sometimes consider ω as a subspace of L ≥ by identifying α ∈ ω with h α, i ∈ L ≥ . The following lemma is well known. Recall that a subset of ω (or L + ) is stationary if it meets any closed and unbounded (abbreviated club ) subset of ω (or L + ).A club subset of ω (or L + ) is stationary (see e.g. [4, Lemma 1.15]) and contains a copy of ω . Lemma 3.2. (a) If U is an open subset of L whose intersection with the diagonal is stationary in ω (thatis, { α ∈ ω : h α, α i ∈ U } is stationary), then U contains [ α, ω ) for some α ∈ ω .(b) If U is an open subset of L + × [0 , whose intersection with the horizontal line L + × { a } isstationary, then U contains [ α, ω ) × ( a − n , a + n ) for some α ∈ ω , n ∈ ω .(c) Let C ⊂ L be closed. If C has both projections unbounded in L + , then it intersectsthe diagonal in a closed and unbounded set. If C has only one projection unbounded, say thehorizontal one, then it intersects some horizontal line L + × { a } in a closed and unbounded set(and ditto for the vertical projection).(d) L + × R , L ≥ × R , L and ( L ≥ ) are normal spaces.Proof. The elementary proofs of (a) to (c) are essentially done in [7, Lemma 3.4 & Example3.5]. We now discuss the normality claims. It is easy to check that since L + = (0 , ∪ [1 , ω )is the union of a Lindel¨of and a countably compact space, it is countably paracompact. ByTheorem 2.2 (c) L + × R is normal. Now, assume that A, B are disjoint closed sets of L . Recallthat the intersection of two (actually, countably many) club subsets of L + is again club [4,Lemma 1.15]. The diagonal is a copy of L + , by (c) one of A, B has one bounded projection, say B ⊂ L + × (0 , α ] for some α ∈ ω ⊂ L + . We finish by using the result for L + × R . The proofsfor L ≥ are almost the same. xample 3.3 (Folklore) . A normal surface with a non-collared boundary.Details.
The closed octant O = {h x, y i ∈ ( L ≥ ) : y ≤ x } has a non collared boundary sinceany open set containing the diagonal is not homeomorphic to the product L ≥ × [0 , O is normal. Example 3.4 (Folklore) . A surface with a collared -set boundary which is not strongly collared.Details. Set S = L + × [0 , − ω × { } . The boundary ( L + − ω ) is an uncountable discrete union of openintervals, is collared (with collar ( L + − ω ) × [0 , h : ( L + − ω ) × [0 , → S be a strong collar and W be its closed image in S . Since( L + − ω ) × [0 ,
1] does not contain a copy of ω , W is bounded in each horizontal line L + × { a } , a ∈ (0 , S − W contains [ β a , ω ) × ( a − n a , a + n a ) for some β a , n a by Lemma 3.2 (b). By Lindel¨ofness, let E ⊂ R be countable such that ∪ a ∈ E ( a − n a , a + n a )covers (0 ,
2) and set β = sup a ∈ E β a . Then S − W ⊃ [ β, ω ) × (0 , β would not be collared.Our next examples are based on procedures to add (or delete) boundary components insurfaces called Pr¨uferisation , Moorisation and
Nyikosisation . They are described in great detailin [4, Chapter 1.3], we will thus only give a geometrical idea of the constructions.Start with a half plane R ≥ × R , and choose a point h , a i on the vertical axis. We replacethis point by a new boundary component R a , which is a copy of R , as follows. A neighborhoodof some point x ∈ R a is given by the union of an interval ( u, v ) ⊂ R a containing x and a“wedge” comprised between the lines of slopes u and v pointing at h , a i and some verticalline { w } × R , w >
0. We then say that we have
Pr¨uferised the half plane at height (or atpoint) a . Figure 2 is a graphical description, mirrored horizontally (to better fit with thenatural reading direction). The new boundary then consists of the old boundary minus {h , a i} together with the new boundary component R a . It takes a moment’s thought to accept thatwe can perform this Pr¨uferisation at each point in the vertical axis and still have a surface(which is the original Pr¨ufer surface, actually), because the added boundary components areindependant of each other, so to say. We can also Pr¨uferise only at some (but not all) pointsand still obtain a surface, as long as we remove the rest of the initial boundary if the set of a where the Pr¨ufersiation is done is not closed. Moreover, the added boundary componentsform a discrete family: The union of R a and a disk tangent to the vertical line at height a is aneighborhood of R a (and intersects only this boundary component).The process of Moorisation at point a consists of first Pr¨uferising at a and then identify x with − x in the added boundary component R a . The points in R a are in a sense pushed inside theinterior of the surface. As before, we can also Moorise at each point at once, or just some ofthem.The Nyikosisation at point a is done similarly, but this time we add a boundary componentwhich is a copy of L + . Instead of taking wedges, we take a family of curves c α ( α ∈ ω ) pointingat h , a i such that if β < α , then close enough to the vertical axis c α is above c β . (Such a familyof 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 [4, Example 1.29] for moredetails.) Then a neighborhood of a point in the added longray is given by an interval in thelongray union the space between two curves (corresponding to the endpoints of the interval)and a vertical line. In particular, c α has α ∈ L + as its unique limit point.Notice that the resulting surface in any of these constructions is separable, as the interior ofthe half plane is dense. Example 3.5 (Rado) . A separable surface with metrisable interior and a non-collared metris-able boundary.Details.
Pr¨uferise the half plane at each height. The resulting surface has then uncountably manyboundary components and thus a non-collared metrisable boundary by Corollary 2.8. Theboundary is made of a discrete uncountable collection of copies of R and is thus metrisable. Example 3.6 (Nyikos) . A surface with a non-collared single boundary component and metris-able interior.Details.
The Nyikosisation (at any point) of the half plane has a non-collared boundary by Theorem 1.2(c).The boundary is a 0-set in the two previous examples by Theorem 1.3 (b). We shall nowgive an example of a manifold where it is not the case. First we need the following fact. Wewere not able to spot a reference for a proof, so we give one.
Lemma 3.7.
Let P be the surface obtained by Pr¨uferising the half plane at each height and let ∂ P = ∪ a ∈ R R a . Let G ⊂ R be a non-meagre set (hence, in particular, somewhere dense). Let U e an open subset of P such that for each a ∈ G , there is some x a ∈ R a ∩ U . Then U containsa ‘strip’ along the boundary (0 , c ) × ( u, v ) (where c, u, v ∈ R , c > , u < v ).Proof. Let q n be a countable dense subset of (0 , π ). Take a point x a in U ∩ R a , for a ∈ G . Foreach such x a , there is sector S k,n,m,a pointing at a , centered on the line making an angle q k withthe horizontal, with interior angle q n and height 1 /m , contained in U . Let G k,n,m = { a ∈ G : S 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 interior angle is fixed at q n and the height fixed at1 /m , U contains a parallelogram of height 1 /m and angle q k . The rest follows.Notice that this lemma also holds if some of the boundary components are Moorised. Example 3.8.
A separable surface with a non -set hence non-strongly collared, but metrisableand collared boundary.Details. Let S be the surface S defined by Pr¨uferising the half plane at each height and then Moorisingthe boundary components R a for a ∈ R − Q . The boundary being ∪ a ∈ Q R a , it is collared byLindel¨ofness and Theorem 1.2. Suppose that f : S → [0 ,
1] witnesses that ∂S is a 0-set. For a ∈ R − Q , the Moorised boundary components R a of P are sent in (0 ,
1] by f . Hence there issome n such that U n = f − (( n , R a for a non-meagre set of a . ByLemma 3.7 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 [ n , R a is in this closure for some rational a , which contradicts the factthat it is a boundary component of S on which f should be 0. Example 3.9.
A separable surface with a non -set hence non-strongly collared, but metrisable,collared and connected boundary.Details. By identifying some points in the boundary in Example 3.8. Let q n , n ∈ ω be an enumerationof Q . If B ⊂ R , we write B q for the corresponding subset of R q (the boundary component atheight q ). We identify now x ∈ (1 , ∞ ) q n with − x ∈ ( −∞ , − q n +1 . The resulting boundary is acopy of R . See Figure 3. Example 3.10 (R.L. Moore, essentially) . A separable surface with uncountably many circleboundary components.Details.
Take the surface of Example 3.5 obtained by Pr¨uferising the half plane at each height. In eachboundary component R a identify x with − x when x ≥
1. (See Figure 3.) A small piece ofeach R 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 showingthat even closed discrete subsets can exhibit a reluctance to act decently. Example 3.11.
The boundary of the surface of Example 3.5 contains a countable closed discretesubset which is not a -set.Details. For each a ∈ Q choose one point in the boundary component R a . This defines a closeddiscrete set which is not a 0-set, the proof being the same as the one given in Example 3.8.However, any closed discrete subset of this surface is a G δ . This is not the case in the nextexample. Example 3.12.
A surface with metrisable non-collared boundary which contains a closed dis-crete subset which is not a G δ (hence not a -set). A variant of this space is cited in [5, Remark 4.2], which is one of the reasons we chosed toinclude it here. Although we could not spot one in the literature, such examples are known fora long time, see for instance [7, Problem 1.2 & Corollary 2.16]. (A normal example is way moreelusive, since none exist under various set-theoretic assumptions, for instance V = L .) Theproof of the claimed properties is based on the well known Pressing down lemma (also knownas Fodor’s Lemma), which can be found in any book about set theory, for instance [6, II.6.15]. Lemma 3.13 (Fodor’s Lemma) . Let S be a stationary subset of ω and f : S → ω be suchthat f ( α ) < α for each α ∈ S . Then there is α such that f − ( { α } ) is stationary.Details. We start with the octant O of Example 3.3, and Pr¨uferise it on the diagonal as follows.The idea is to add new boundary components (real lines) at points h α, α i for α ∈ ω limit.What happens to the rest of the boundary is irrelevant hence me might as well delete it. Weadd these components in such a way that the vertical line { α } × [0 , α ) converges to the 0point of the real line attached at h α, α i . We might do it as follows. For each limit α ∈ ω ,choose a strictly increasing sequence α n ∈ L + converging to it. In what follows, the intervals(0 , α ) , (0 , α ] are understood as being in L + if α ≥ ω , otherwise they are intervals of R . Fix ahomeomorphism Φ : (0 , α ] → (0 ,
1] sending α n to 1 − /n , and extend it to an homeomorphism(0 , α + 1) → (0 ,
2) the obvious way. This yields a homeomorphism Ψ : (0 , α + 1) → (0 , {h x, y i ∈ (0 , : y ≤ x } , Pr¨uferise at h , i such that the 0 point of the added real lineis the limit of the vertical line below h , i , and pull back this Pr¨uferisation in the octant byΨ − . Figure 4 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 componentadded at h α, α i . Figure 5: Double Fodor
Then D = { α : α ∈ ω , α limit } is a closed discrete subset of S : the intersection of theneighborhood depicted in Figure 4 with D is { α } . We now prove that D is not a G δ by showingthat for any open set U containing D , there is β ∈ ω such that for each γ ≥ β , U ⊃ [ ζ, ω ) × { γ } for some ζ < ω . 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 , 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 lemma3.13, there is β < ω such that β ( α ) = β for a stationary set E ⊂ ω of α . This means that ifwe take γ ≥ β , U intersects the horizontal line at height γ on a stationary set, more preciselythat for each α ∈ E , α > β , there is ζ ( α ) < α such that [ ζ ( α ) , α ] × { γ } ⊂ U . Applying Fodoragain, we see that U ⊃ [ ζ, ω ) × { γ } for some ζ < ω , which is what we wanted to prove.Those who enjoy pictures may consult Figure 5 for a summary of this double application ofFodor’s lemma.Our last example is not a manifold but shows that Lemma 2.4 does not hold when the spaceis not assumed to be regular. Example 3.14 (Folklore) . A Hausdorff space X with a closed locally collared subset B whichis not locally strongly collared. h , , i in Example 3.14 and its closure. Details.
Let H = [0 , be given the half disk topology [8, Example 78], that is, neighborhoodsof points in [0 , × (0 ,
1) are usual open sets in the plane and neighboorhoods of p = h x, i arethe union of { p } and an open disk centered at p (or any open neighborhood of p in the plane)intersected with [0 , × (0 , H is a Hausdorff non-regular space. Let X be H × [0 , E = (0 , × { } × { } removed. Set B to be the subsetof points with 0 third coefficient. Then B is closed and B × [0 , ⊂ X is an open collar. Sincewe removed E , the topology on B is the usual topology as a subset of the plane, in particularthe product of any of its subsets with [0 ,
1) is regular. (Recall that any product of regular spacesis regular, see e.g. in [3, Theorem 2.3.11].) But any closed neighborhood of x = h , , i ∈ B is not regular because, as seen in Figure 6, it will contain a piece of a line [0 , a ] × { } × { c } for some a, c . Hence such a closed neighborhood cannot be homeomorphic to U × [0 ,
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