topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
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.
The coproduct in of a small family of paracompacta is a paracompactum.
The proof is very easy. See here.
A closed subspace of a paracompactum is also a paracompactum.
Let be an open cover of ; note is the maximal open in such that . The cover . If is a locally finite open refinement of this cover (via paracompactness of ), then the family is a locally finite open refinement of the family of sets . Finally, is Hausdorff because Hausdorffness is a hereditary property.
In , the pushout of a (closed/open) embedding along any continuous map ,
is again a (closed/open) embedding.
For the sake of convenience, we reproduce the proof given here.
Since is faithful, we have that monos are reflected by ; also monos and pushouts are preserved by since has both a left adjoint and a right adjoint. In , the pushout of a mono along any map is a mono, so we conclude is monic in . Furthermore, such a pushout diagram in is also a pullback, so that we have the Beck-Chevalley equality (where is the direct image map between power sets, and is the inverse image map).
To prove that is a subspace, let be any open set. Then there exists open such that because is a subspace inclusion. If and 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 which extends the pair of maps . It follows that , so that is a subspace inclusion.
If moreover is an open inclusion, then for any open we have that (since is monic) and (by Beck-Chevalley) is open in . By the definition of the topology on , it follows that is open, so that is an open inclusion. The same proof, replacing the word “open” with the word “closed” throughout, shows that the pushout of a closed inclusion is a closed inclusion .
(See also Michael's theorems.)
It is well-known (see here) that a space is a paracompactum iff every open cover of 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 (singletons are closed); see separation property.
Recall that a relation in a category is a jointly monic span
and that the relation is entire if is an epimorphism. We will be working in the category , where epimorphisms are the same as surjective continuous functions. A selection for such a relation is by definition a section of , where of course in this means a continuous section. (In the category , 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 is a space, let denote its topology, considered as a subset of the power set . Adapting the terminology of Michael, we will say a relation is lsc (“lower semi-continuous”) if the composite
restricts to a map . (In more down-to-earth terms, if we consider as a subspace of , this says the image under the projection is open in whenever is open in .)
Let be a Banach space (for an example relevant to paracompactness concerns, could be , the Banach space freely generated by a set that indexes an open cover of a given space ). We will say an (entire) relation from to in is closed and convex if for each the fiber is a closed convex subset of .
(E. Michael’s selection theorem) A space is a paracompactum iff the condition
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.
If is entire, lsc, closed and convex, and is a continuous map, then the pullback relation (as in the pullback diagram)
is also entire, lsc, closed and convex.
Clearly is surjective (being a pullback of a surjection), so the relation is entire, and for we have so that is closed and convex if is. We also have
where the second equation follows from a Beck-Chevalley equation (induced by the pullback square). The map preserves openness since is lsc, and preserves openness by continuity, so preserves openness, which proves is lsc.
The next result is an important observation about extending continuous selections.
If satisfies the condition of Theorem and is a closed embedding, then for every , every selection of can be extended to a selection of .
Replace the relation by the relation from to where if , and if . Clearly every fiber is closed, convex, and inhabited, so is closed and convex and entire. We check that is lsc. Let be open in ; we must verify that every point of is an interior point of that set. If , this is clear since
is open. If , then by continuity of there is open neighborhood of such that for all , and then one may easily check that
so that the left side is an open neighborhood of contained within the right side. Thus is lsc.
Since hypothesis holds for , we conclude that admits a selection ; by construction of we must have that , and this completes the proof.
Now we put Michael’s selection theorem to use in developing colimits of paracompacta.
If are paracompacta and is a closed embedding and is a continuous map, then in the pushout diagram in
the space is a paracompactum (and is a closed embedding, by Lemma ).
(Compare a similar result for normal spaces, here.)
Let be an entire, lsc, closed and convex relation to a Banach space ; by Michael’s selection criterion, it suffices to show that is and admits a section. That is is easy: a point either belongs to the closed subspace or it doesn’t. If it does, then because is Hausdorff, it is a closed point in a closed subspace, so it is a closed point in . If it doesn’t, then and is a singleton in and thus closed in since is Hausdorff, i.e., the inverse image of under the quotient map is the closed subset , so must be closed in by definition of quotient topology.
Thus we have only to prove admits a section. Using Lemma , we pull back to entire, lsc, closed and convex relations to from and from :
By paracompactness of and the selection criterion, there is a section of . The composite in turn induces a selection . By Lemma , the selection may be extended to a selection . The two maps
may be pasted together (i.e., their respective composites with and agree) to give a map out of the pushout, , which is easily checked to be a section of .
If is a countable sequence of closed embeddings between paracompacta, then the colimit is also a paracompactum.
Clearly is : for any , the intersection is closed in for all , so must a closed point of .
Let denote a component of the colimit cocone. Let from to a Banach space be an entire, lsc, closed and convex relation. Pulling this back to , by paracompactness we have a selection . Given a selection , we may extend along to a selection using paracompactness and Lemma . These selections, being compatible with the inclusions , paste together to give a selection .
In particular this may be used to see that CW-complexes are paracompact Hausdorff spaces.
Last revised on June 6, 2017 at 09:08:01. See the history of this page for a list of all contributions to it.