nLab
colimits of paracompact Hausdorff spaces

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Limits and colimits

Contents

Introduction

In this article we collect some results on colimits of paracompact Hausdorff spaces as computed in the category of topological spaces Top, and particular conditions under which the colimit is again paracompact and Hausdorff.

The account is designed to be parallel to that given in colimits of normal spaces, centering particularly on how paracompactness may be reformulated in terms of an extension or selection property. The relevant result, due to Ernest Michael, is particularly well-adapted to the study of colimits.

In the sequel, a paracompactum (pl. paracompacta) is a paracompact Hausdorff space, just as compactum is a compact Hausdorff space.

Basic results

Proposition

The coproduct in TopTop of a small family of paracompacta is a paracompactum.

The proof is very easy. See here.

Proposition

A closed subspace AA of a paracompactum XX is also a paracompactum.

Proof

Let U αU_\alpha be an open cover of AA; note ¬AU α\neg A \cup U_\alpha is the maximal open V αV_\alpha in XX such that U α=V αAU_\alpha = V_\alpha \cap A. The V αV_\alpha cover XX. If 𝒱\mathcal{V} is a locally finite open refinement of this cover (via paracompactness of XX), then the family {VA:V𝒱}\{V \cap A: V \in \mathcal{V}\} is a locally finite open refinement of the family of sets U αU_\alpha. Finally, AA is Hausdorff because Hausdorffness is a hereditary property.

Lemma

In TopTop, the pushout jj of a (closed/open) embedding ii along any continuous map ff,

A i B f po g C j D,\array{ A & \stackrel{i}{\hookrightarrow} & B \\ \mathllap{f} \downarrow & po & \downarrow \mathrlap{g} \\ C & \underset{j}{\hookrightarrow} & D, }

is again a (closed/open) embedding.

Proof

For the sake of convenience, we reproduce the proof given here.

Since U=hom(1,):TopSetU = \hom(1, -): Top \to Set is faithful, we have that monos are reflected by UU; also monos and pushouts are preserved by UU since UU has both a left adjoint and a right adjoint. In SetSet, the pushout of a mono along any map is a mono, so we conclude jj is monic in TopTop. Furthermore, such a pushout diagram in SetSet is also a pullback, so that we have the Beck-Chevalley equality if *=g * j:P(C)P(B)\exists_i \circ f^\ast = g^\ast \exists_j \colon P(C) \to P(B) (where i:P(A)P(B)\exists_i \colon P(A) \to P(B) is the direct image map between power sets, and f *:P(C)P(A)f^\ast: P(C) \to P(A) is the inverse image map).

To prove that jj is a subspace, let UCU \subseteq C be any open set. Then there exists open VBV \subseteq B such that i *(V)=f *(U)i^\ast(V) = f^\ast(U) because ii is a subspace inclusion. If χ U:C2\chi_U \colon C \to \mathbf{2} and χ V:B2\chi_V \colon B \to \mathbf{2} are the maps to Sierpinski space that classify these open sets, then by the universal property of the pushout, there exists a unique continuous map χ W:D2\chi_W \colon D \to \mathbf{2} which extends the pair of maps χ U,χ V\chi_U, \chi_V. It follows that j 1(W)=Uj^{-1}(W) = U, so that jj is a subspace inclusion.

If moreover ii is an open inclusion, then for any open UCU \subseteq C we have that j *( j(U))=Uj^\ast(\exists_j(U)) = U (since jj is monic) and (by Beck-Chevalley) g *( j(U))= i(f *(U))g^\ast(\exists_j(U)) = \exists_i(f^\ast(U)) is open in BB. By the definition of the topology on DD, it follows that j(U)\exists_j(U) is open, so that jj is an open inclusion. The same proof, replacing the word “open” with the word “closed” throughout, shows that the pushout of a closed inclusion ii is a closed inclusion jj.

Michael’s selection characterization of paracompactness

(See also Michael's theorems.)

It is well-known (see here) that a T 1T_1 space XX is a paracompactum iff every open cover of XX admits a subordinate partition of unity. Michael’s innovation was in working out how the partition of unity characterization may be recast in terms of existence of continuous sections of suitable projection maps.

From here on out, all spaces will be assumed to be T 1T_1 (singletons are closed); see separation property.

Recall that a relation in a category is a jointly monic span

R p q X Y\array{ & & R & & \\ & \mathllap{p} \swarrow & & \searrow \mathrlap{q} & \\ X & & & & Y }

and that the relation is entire if pp is an epimorphism. We will be working in the category TopTop, where epimorphisms are the same as surjective continuous functions. A selection for such a relation is by definition a section of pp, where of course in TopTop this means a continuous section. (In the category SetSet, the axiom of choice may be equivalently stated as saying that every entire relation admits a selection, a formulation which is sometimes credited to Peirce.)

If XX is a space, let 𝒪X\mathcal{O} X denote its topology, considered as a subset of the power set PXP X. Adapting the terminology of Michael, we will say a relation XpRqYX \stackrel{p}{\leftarrow} R \stackrel{q}{\to} Y is lsc (“lower semi-continuous”) if the composite

PYq *PR pPXP Y \stackrel{q^\ast}{\to} P R \stackrel{\exists_p}{\to} P X

restricts to a map pq *:𝒪Y𝒪X\exists_p q^\ast: \mathcal{O} Y \to \mathcal{O} X. (In more down-to-earth terms, if we consider RR as a subspace of X×YX \times Y, this says the image π X(R(X×V))\pi_X(R \cap (X \times V)) under the projection π X:X×YX\pi_X: X \times Y \to X is open in XX whenever VV is open in YY.)

Let BB be a Banach space (for an example relevant to paracompactness concerns, BB could be L 1(S)L^1(S), the Banach space freely generated by a set SS that indexes an open cover {U s} sS\{U_s\}_{s \in S} of a given space XX). We will say an (entire) relation (R,p,q)(R, p, q) from XX to BB in TopTop is closed and convex if for each xXx \in X the fiber R x= qp *(x)R_x = \exists_q p^\ast(x) is a closed convex subset of BB.

Theorem

(E. Michael’s selection theorem) A T 1T_1 space XX is a paracompactum iff the condition

  • for every Banach space BB, every entire, lsc, closed and convex relation (R,p,q)(R, p, q) from XX to BB admits a selection

is satisfied.

For now we omit giving the proof (given in Michael).

Before applying this theorem, we need a few lemmas that play supporting roles.

Lemma

If (R,p,q):XB(R, p, q): X \nrightarrow B is entire, lsc, closed and convex, and f:AXf: A \to X is a continuous map, then the pullback relation (f *R,r,qg):AY(f^\ast R, r, q g): A \nrightarrow Y (as in the pullback diagram)

f *R g R r (pb) p q A f X B\array{ & & f^\ast R & \stackrel{g}{\to} & R & & \\ & \mathllap{r} \swarrow & (pb) & \swarrow \mathrlap{p} & & \searrow \mathrlap{q} & \\ A & \underset{f}{\to} & X & & & & B }

is also entire, lsc, closed and convex.

Proof

Clearly rr is surjective (being a pullback of a surjection), so the relation f *Rf^\ast R is entire, and for aAa \in A we have (f *R) a=R f(a)(f^\ast R)_a = R_{f(a)} so that f *Rf^\ast R is closed and convex if RR is. We also have

r(qg) *= rg *q *=f * pq *\exists_r (q g)^\ast = \exists_r g^\ast q^\ast = f^\ast \exists_p q^\ast

where the second equation follows from a Beck-Chevalley equation (induced by the pullback square). The map pq *\exists_p q^\ast preserves openness since RR is lsc, and f *f^\ast preserves openness by continuity, so r(qg) *\exists_r (q g)^\ast preserves openness, which proves f *Rf^\ast R is lsc.

The next result is an important observation about extending continuous selections.

Lemma

If XX satisfies the condition \bullet of Theorem and f:AXf: A \hookrightarrow X is a closed embedding, then for every (R,p,q):XB(R, p, q): X \to B, every selection σ:Af *R\sigma: A \to f^\ast R of (f *R,r,qg)(f^\ast R, r, q g) can be extended to a selection of (R,p,q)(R, p, q).

Proof

Replace the relation RR by the relation SS from XX to BB where S f(x)={σ(x)}S_{f(x)} = \{\sigma(x)\} if xAx \in A, and S x=R xS_x = R_x if xf(A)x \notin f(A). Clearly every fiber S xS_x is closed, convex, and inhabited, so SS is closed and convex and entire. We check that SS is lsc. Let VV be open in YY; we must verify that every point xx of π X(S(X×V))\pi_X(S \cap (X \times V)) is an interior point of that set. If x¬Ax \in \neg A, this is clear since

¬Aπ X(S(X×V))=¬Aπ X(R(X×V)\neg A \cap \pi_X(S \cap (X \times V)) = \neg A \cap \pi_X(R \cap (X \times V)

is open. If xAx \in A, then by continuity of σ\sigma there is open neighborhood UU of xx such that σ(x)V\sigma(x') \in V for all xUAx' \in U \cap A, and then one may easily check that

Uπ X(R(X×V))π X(S(X×V))U \cap \pi_X(R \cap (X \times V)) \subseteq \pi_X(S \cap (X \times V))

so that the left side is an open neighborhood of xx contained within the right side. Thus SS is lsc.

Since hypothesis \bullet holds for XX, we conclude that SS admits a selection τ\tau; by construction of SS we must have that τ| A=σ\tau |_A = \sigma, and this completes the proof.

Applications to colimits of paracompacta

Now we put Michael’s selection theorem to use in developing colimits of paracompacta.

Theorem

If X,Y,ZX, Y, Z are paracompacta and h:XZh: X \to Z is a closed embedding and f:XYf: X \to Y is a continuous map, then in the pushout diagram in TopTop

X h Z f g Y k W,\array{ X & \stackrel{h}{\to} & Z \\ \mathllap{f} \downarrow & & \downarrow \mathrlap{g} \\ Y & \underset{k}{\to} & W, }

the space WW is a paracompactum (and k:YWk: Y \to W is a closed embedding, by Lemma ).

(Compare a similar result for normal spaces, here.)

Proof

Let WpRqBW \stackrel{p}{\leftarrow} R \stackrel{q}{\to} B be an entire, lsc, closed and convex relation to a Banach space BB; by Michael’s selection criterion, it suffices to show that WW is T 1T_1 and pp admits a section. That WW is T 1T_1 is easy: a point xWx \in W either belongs to the closed subspace YWY \hookrightarrow W or it doesn’t. If it does, then because YY is Hausdorff, it is a closed point in a closed subspace, so it is a closed point in WW. If it doesn’t, then k 1(x)=k^{-1}(x) = \emptyset and g 1(x)g^{-1}(x) is a singleton in ZZ and thus closed in ZZ since ZZ is Hausdorff, i.e., the inverse image of xx under the quotient map (k,g):Y+ZW(k, g): Y + Z \to W is the closed subset +g 1(x)\emptyset + g^{-1}(x), so xx must be closed in WW by definition of quotient topology.

Thus we have only to prove pp admits a section. Using Lemma , we pull back (R,p,q)(R, p, q) to entire, lsc, closed and convex relations to BB from YY and from ZZ:

X h Z f g Y k W g *R p k *R R.\array{ X & \stackrel{h}{\to} & Z & & \\ \mathllap{f} \downarrow & & \downarrow \mathrlap{g} & \nwarrow & \\ Y & \underset{k}{\to} & W & & g^\ast R \\ & \nwarrow & & \nwarrow \mathrlap{p} & \downarrow \\ & & k^\ast R & \to & R. }

By paracompactness of YY and the selection criterion, there is a section ss of k *RYk^\ast R \to Y. The composite sf:Xk *Rs f: X \to k^\ast R in turn induces a selection t:Xf *k *Rh *g *Rt: X \to f^\ast k^\ast R \cong h^\ast g^\ast R. By Lemma , the selection tt may be extended to a selection u:Zg *Ru: Z \to g^\ast R. The two maps

Ysk *RR,Zug *RRY \stackrel{s}{\to} k^\ast R \to R, \qquad Z \stackrel{u}{\to} g^\ast R \to R

may be pasted together (i.e., their respective composites with ff and hh agree) to give a map out of the pushout, i:WRi: W \to R, which is easily checked to be a section of pp.

Proposition

If (i n:X nX n+1) n(i_n: X_n \to X_{n+1})_{n \in \mathbb{N}} is a countable sequence of closed embeddings between paracompacta, then the colimit X=colim nX nX = colim_n X_n is also a paracompactum.

Proof

Clearly XX is T 1T_1: for any xXx \in X, the intersection {x}X n\{x\} \cap X_n is closed in X nX_n for all nn, so xx must a closed point of XX.

Let j n:X nXj_n: X_n \to X denote a component of the colimit cocone. Let (R,p,q)(R, p, q) from XX to a Banach space BB be an entire, lsc, closed and convex relation. Pulling this back to X 0X_0, by paracompactness we have a selection X 0j 0 *RX_0 \to j_0^\ast R. Given a selection X nj n *RX_n \to j_n^\ast R, we may extend along i ni_n to a selection X n+1j n+1 *RX_{n+1} \to j_{n+1}^\ast R using paracompactness and Lemma . These selections, being compatible with the inclusions i ni_n, paste together to give a selection XRX \to R.

In particular this may be used to see that CW-complexes are paracompact Hausdorff spaces.

References

  • Ernest Michael, Continuous Selections I, The Annals of Mathematics, 2nd Series, Vol. 63 No. 2. (March 1956), 361-382. stable URL, (pdf)

Last revised on June 6, 2017 at 05:08:01. See the history of this page for a list of all contributions to it.