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
Recall:
Let $(X,\tau)$ be a topological space.
An open cover $\{U_i \subset X\}_{i \in I}$ of $X$ is called locally finite if for all point $x \in X$, there exists a neighbourhood $U_x \supset \{x\}$ such that it intersects only finitely many elements of the cover, hence such that $U_x \cap U_i \neq \emptyset$ for only a finite number of $i \in I$.
(refinement of open covers)
Let $(X,\tau)$ be a topological space, and let $\{U_i \subset X\}_{i \in I}$ be a open cover.
Then a refinement of this open cover is a set of open subsets $\{V_j \subset X\}_{j \in J}$ which is still an open cover in itself and such that for each $j \in J$ there exists an $i \in I$ with $V_j \subset U_i$.
Now:
(paracompact topological space)
A topological space $(X,\tau)$ is called paracompact if every open cover of $X$ has a refinement (def. 2) by a locally finite open cover (def. 1).
(differing terminology)
As with the concept of compact topological spaces (this remark), some authors demand a paracompact space to also be a Hausdorff topological space. See at paracompact Hausdorff space.
Every compact space is paracompact.
Every locally connected locally compact topological group is paracompact (this prop.).
locally compact and second-countable Hausdorff space are paracompact.
(Euclidean space is paracompact)
For $n \in \mathbb{N}$, then the Euclidean space $\mathbb{R}^n$, regarded with its metric topology is paracompact.
Euclidean space is evidently locally compact and sigma-compact. Therefore the statement follows since locally compact and sigma-compact spaces are paracompact (prop. 2).
Paracompactness is preserved by forming disjoint union spaces (coproducts in Top).
Consider a disjoint union $X = \coprod X_\lambda$ whose components are paracompact. As the union is disjoint, the components, that is to say, the $X_\lambda$, are open in $X$. Thus any open cover, say $\mathcal{U}$, of $X$ has a refinement by open sets, say $\mathcal{V}$, such that each $V \in \mathcal{V}$ is contained in some $X_\lambda$. Thus we can write $\mathcal{V} = \coprod \mathcal{V}_\lambda$. As each $X_\lambda$ is paracompact, each $\mathcal{V}_\lambda$ has a locally finite refinement, say $\mathcal{W}_\lambda$. Then let $\mathcal{W} := \coprod \mathcal{W}_\lambda$. As each $\mathcal{W}_\lambda$ is a refinement of the corresponding $\mathcal{V}_\lambda$, $\mathcal{W}$ is a refinement of $\mathcal{V}$, and hence of $\mathcal{U}$. As each point of $X$ has a neighbourhood which meets only elements of one of the $\mathcal{W}_\lambda$, and as that $\mathcal{W}_\lambda$ is locally finite, $\mathcal{W}$ is locally finite. Thus $\mathcal{U}$ has a locally finite refinement.
finite-dimensional manifolds are locally compact, so we have the results above, but we also have some converses:
a finite-dimensional Hausdorff topological manifold is paracompact precisely if it is metrizable
a finite-dimensional Hausdorff topological manifold is paracompact precisely if each component is second-countable
infinite-dimensional manifolds are generally not locally compact, but we still have some results:
The Frechet smooth loop space of a compact finite dimensional manifold is paracompact.
More generally, if $E$ is the sequential limit of separable Hilbert spaces $H_n$, such that the canonical projections
satisfy
for any open ball $B$ in $H_n$, then $E$ is paracompact, and furthermore admits smooth partitions of unity.
Urs Schreiber: don’t we need some extra assumption here? Otherwise why wouldn’t this imply that every space modeled on $\mathbb{R}^n$ is paracompact, while it is only the second-countable such that are?
Toby: Probably Brylinski has a requirement of metrisability or something.
Andrew Stacey: More generally, the coproduct of paracompact spaces is again paracompact.
Spivak, in A Comprehensive Introduction to Differential Geometry, I has an appendix on non-metrisable manifolds. He starts (in the appendix) by defining a manifold to be a Hausdorff locally Euclidean space. Then he proves that for any manifold $M$ TFAE:
The bit to highlight is the words “Each component of …” in the first two.
Toby: Right, and I've got the ‘each component’ clause in the section on finite-dimensional manifolds. But how does this work for infinite-dimensional manifolds? (I'm also unsure how things work for non-Hausdorff manifolds, or more generally for non-Hausdorff locally compact spaces.
David Roberts: Lang has the result that second countable manifolds are paracompact. For him manifold seems to mean that the space is Hausdorff and locally euclidean, with no restriction on the type of vector space. Further, smooth partitions of unity exist. This is Corollary 3.4 in the 2002 edition of Introduction to differentiable manifolds
Andrew Stacey Surely “locally Euclidean” implies that the model space is $\mathbb{R}^n$. Is this the book where he deals with finite and infinite dimensional manifolds all in one go? In that case, I would caution that if I remember right (don’t have the book in front of me) he’s using manifolds modelled on at most Banach spaces. For smooth partitions of unity to exist then you need to know that there is a smooth bump function, which hinges on some properties of the norm. Kriegl and Michor (who else?) address this in chapter III of their book. In particular, they quote a result due to Kurzweil (1954) that $C([0,1])$ and $\ell^1$ are not $C^1$-regular.
To Toby, I guess that the issue about components for infinite dimensional manifolds is dealt with in what I did before. The question reduces to figuring out if a specific component is paracompact. To use “metrisable implies paracompact” you’ve got to be on a Frechet manifold (since Frechet is the limit of metrisability). Brylinski’s construction outlined about is sufficient to ensure that (countable family of semi-norms). To have smooth partitions of unity, you then need smoothly regular. If the semi-norms are smooth (away from zero) then, obviously, that’s sufficient. In Brylinski’s construction then that comes from the fact that the semi-norms are defined by Hilbertian norms. I’m not sure what the condition on the closures is for, can anyone scan through the proof and see why they are used? For the proof of paracompactness and partitions of unity then I don’t see immediately why they are needed. I expect I’m being dense? but if someone could quickly enlighten me, I’d be grateful.
(Incidentally, since this query box is contained within a list, it’s important that all paragraphs are indented properly, otherwise strange things happen)
David Roberts: When I said locally Euclidean, I was being lazy. I meant modelled on some vector space. And yes Lang does finite and infinite dimensions at the same time. Regarding the condition Brylinski uses, he says it implies that the image of $p_n : E \to H_n$ from the ILH space $E$ to each of the Hilbert spaces $H_n$ in the sequence is dense. I don’t know how this is used, I’m relying on G00gle books. Ah, but now that I check, Lang also says unless specified, vector space will mean finite dimensional vector space …
Andrew Stacey Hmm, I guess I’ll have to get Lang and Brylinski out of the library to see exactly what’s going on here. I don’t see how density can make any difference since we could always restrict to the closure of $p_n(E)$ in $H_n$.
CW-complexes are paracompact Hausdorff spaces (Miyazaki 52), see for instance Hatcher, appendix of section 1.2. For a discussion that highlights which choice principles are involved, see (Fritsch-Piccinini 90, Theorem 1.3.5 (p. 29 and following)).
metric spaces
every separable metric space is paracompact;
every metric space whatsoever is paracompact, assuming the axiom of choice; see at metric spaces are paracompact
pseudometric spaces are paracompact under the same conditions, if one does not require Hausdorffness;
In particular we have the following implications
second-countable space and regular space $\Rightarrow$ metrizable space $\Rightarrow$ paracompact space
(the first is Urysohn’s metrization theorem, the second is due to Stone 48, see also at second-countable regular spaces are paracompact and metric spaces are paracompact)
paracompact space and locally metric space $\Rightarrow$ metrizable space
(this is due to Smirnov)
special cases
counterexamples
the long line is not paracompact, even though it is a manifold (unless one specifically requires paracompactness of manifolds) but it fails to be second-countable (even though it is connected) or metrisable.
the Sorgenfrey plane (a product of two Sorgenfrey lines) is not paracompact. This shows that the product of paracompact spaces need not be paracompact.
Dieudonne’s theorem: paracompact Hausdorff spaces are normal
every paracompact finite-dimensional manifold has a partition of unity
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
Care should be taken as to which category one constructs partitions of unity on paracompact spaces. For example, analytic partitions of unity generally do not exist on smooth (finite dimensional) manifolds, even when smooth ones do.
For paracompact spaces, numerable open covers are cofinal in all open covers (in $Top$).
Let $X$ be a paracompact Hausdorff space, and let $\{U_i \subset X\}_{i \in I}$ be an open cover. Then there exists a countable cover
such that each element $V_n$ is a union of open subsets of $X$ each of which is contained in at least one of the elements $U_i$ of the original cover.
(e.g. Hatcher, lemma 1.21)
Let $\{f_i \colon X \to [0,1]\}_{i \in I}$ be a partition of unity subordinate to the original cover, which exists since paracompact Hausdorff spaces equivalently admit subordinate partitions of unity.
For $J \subset I$ a finite set, let
By local finiteness there are only a finite number of $f_k(x)$ greater than zero, hence the condition on the right is a finite number of strict inequalities. Since the $f_i$ are continuous, this implies that $V_J$ is an open subset.
Moreover, $V_J$ is contained in $supp(f_j)$ for $j \in J$ and hence in one of the $U_i$.
Now for $n \in \mathbb{N}$ take
to be the union of the $V_J$ over all subset $J$ with precisely $n$ elements.
The set $\{V_n \subset X\}_{n \in \mathbb{N}}$ is a cover because for any $x \in X$ we have $x \in V_{J_x}$ for
(which is finite by local finitness of the partition of unity).
See at colimits of paracompact Hausdorff spaces.
On paracompact spaces, abelian Čech cohomology does compute abelian sheaf cohomology,
i.e. the canonical morphism $\check{H}(X,A) \to H(X,A)$ for $A$ any chain complex of sheaves is an isomorphism when the topological space underlying $X$ is paracompact.
On a paracompact space $X$, every hypercover of finite height is refined by the Cech nerve of an ordinary open cover.
For a hypercover of height $n \in \mathbb{N}$, this follows by intersecting the open covers that are produced by the following lemma for $0 \leq k \leq n$.
For $X$ a paracompact topological space, let $\{U_\alpha\}_{\alpha \in A}$ be an open cover, and let each $(k+1)$-fold intersection $U_{\alpha_0, \cdots, \alpha_{k}}$ be equipped itself with an open cover $\{V^{\alpha_0, \cdots, \alpha_k}_{\beta}\}$.
Then there exists a refinement $\{U'_{\alpha'}\}$ of the original cover, such that each $(k+1)$-fold intersection $U'_{\alpha'_0, \cdots, \alpha'_k}$ for all indices distinct is contained in one of the $V_\beta$.
This appears as (HTT, lemma 7.2.3.5).
compact topological space, countably compact topological space, locally compact topological space, strongly compact topological space, sequentially compact topological space
The notion of paracompact space was introduced in
That fully normal spaces are equivalently paracompact is due to
General accounts include
R. Engelking, General topology, chapter 5 is dedicated to paracompact spaces
Brian Conrad, Paracompactness and local compactness, pdf
D. K. Burke, Covering properties, in: K. Kunen, J.E. Vaughan (eds.), Handbook of Set-Theoretic Topology, North-Holland (1984) Ch. 9, 347–422
Alan Hatcher, section 1.2 of Vector bundles & K-theory (web)
English Wikipedia: paracompact space
Springer eom: paracompact space, paracompactness criteria
Some properties of paracompact spaces are listed and proven in http://www.helsinki.fi/~hjkjunni/top9.pdf
A basic discussion with an eye towards abelian sheaf cohomology and abelian Čech cohomology is around page 32 of
Discussion of paracompactness of CW-complexes includes