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
A numerable open cover (alias normal cover) is an open cover of a topological space that admits a subordinate partition of unity.
Let $\{U_i\}$ be an open cover of the topological space $X$ (actually Dold doesn’t always require open, see discussion below). It is said to be numerable if there is a collection of continuous functions $\phi_i \colon X \to [0,1]$ (to the closed interval) such that:
the support of the $i$th function is contained in the $i$th subset:
$Supp(\phi_i) \subset U_i$;
at each point $x\in X$, only a finite number of the $\phi_i$ are non-zero;
$\underset{x \in X}{\forall} \sum_i \phi_i(x) \equiv 1$.
(The functions $\{\phi_i\}$ are a partition of unity.)
The collection of preimages $\phi_i^{-1}(0,1]$ then constitute a locally finite cover that refines $\{U_i\}$.
(Ernest Michael, Kiiti Morita, Arthur H. Stone, see Morita \cite[Theorem 1.2]{Morita}.)
For an open cover $\{U_i\}_{i\in I}$ of a topological space $X$ the following properties are equivalent.
$U$ is a numerable cover, i.e., it admits a compatible positive partition;
(Michael \cite[Proposition 2]{Michael}.) $U$ admits a subordinate locally finite partition of unity;
(Stone \cite[Theorems 1 and 2]{Stone}.) $U$ is a normal cover, meaning there is a sequence $W_0=U$, $W_1$, $W_2$, … of open covers of $X$ such that $W_{n+1}$ is a star refinement? of $W_n$ for all $n\ge0$;
$U$ can be refined by the inverse image of an open cover of $Y$ under some continuous map $X\to Y$, where $Y$ is a metrizable topological space;
(Mardešić and Segal \cite[Lemma I.6.1]{MardesicSegal}.) $U$ can be refined by the inverse image of an open cover of $Y$ under some continuous map $X\to Y$, where $Y$ is an absolute neighborhood retract, i.e., a metrizable topological space $Y$ such that any closed embedding $Y\to Z$ into a metrizable topological space $Z$ factors through an open subset $U\subset Z$ such that there is a map $U\to Y$ for which the composition $Y\to U\to Y$ is identity;
$U$ can be refined by a locally finite normal open cover;
$U$ can be refined by a locally finite open cover consisting of cozero sets;
(Michael \cite[Theorem 1]{Michael}, Morita \cite[Theorem 1.2]{Morita2}.) $U$ can be refined by an open cover given by the union of a countable collection of locally finite families of cozero sets.
(Michael \cite[Proposition 1]{Michael}, Hoshina \cite[Theorem 1.1 and 1.2]{Hoshina}.) $U$ can be refined by an open cover given by the union of a countable collection of discrete families of cozero sets.
Numerable open covers form a site called the numerable site. More precisely, numerable open covers are a coverage on the category Top of topological spaces (this is essentially given in Dold’s lectures, A.2.17, but not using this terminology).
For paracompact topological spaces, numerable covers are cofinal in open covers, so that the numerable site is equivalent to the open cover site for such spaces.
There is also a 1944 result by Dieudonnne that numerable covers are cofinal in locally finite covers of normal spaces — need to add this! See, eg, Theorem 6.3 of Howes’ Modern analysis and topology.
Many classical theorems concerning fiber bundles are stated for the numerable site. For example, the classifying space $\mathcal{B}G$ actually classifies bundles which trivialise over a numerable cover. (References? Dold for Milnor's classifying space, and tom Dieck I think for Segal’s) These are called numerable bundles. This is because the standard constructions of the universal bundle by Minor and Segal both are trivialisable over a numerable cover.
Arthur H. Stone?,
Paracompactness and product spaces, Bulletin of the American Mathematical Society 54:10 (1948), 977–983. doi:10.1090/s0002-9904-1948-09118-2.
A note on paracompact spaces. Proceedings of the American Mathematical Society 4:5 (1953), 831–838. doi:10.1090/s0002-9939-1953-0056905-8.
Paracompactness and product spaces. Fundamenta Mathematicae 50:3 (1962), 223–236. doi:10.4064/fm-50-3-223-236.
Products of normal spaces with metric spaces, Mathematische Annalen 154:4 (1964), 365–382. doi:10.1007/bf01362570.
Albrecht Dold, Partitions of unity in the theory of fibrations, Ann. of Math. (2) 78 1963 223–255 MR0155330 jstor doi
Shape Theory. North-Holland Mathematical Library 26 (1982).
Takao Hoshina, Extensions of mappings II. Topics in General Topology, Elsevier (1989).
The appendix of
talks about “stacked covers”: these are useful for ‘decomposing’ numerable covers of products to a sort of parameterised version depending on a numerable cover of the first factor. This is important in looking at concordance of numerable bundles.
Last revised on March 22, 2021 at 10:00:45. See the history of this page for a list of all contributions to it.