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 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 functions $\phi_i:X \to [0,1]$ such that
The open cover $\phi_i^{-1}(0,1]$ is then a locally finite cover that refines $\{U_i\}$. The functions $\{\phi_i\}$ are a partition of unity.
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 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 some result by Bourbaki that I have to look up that numerable covers are cofinal in locally finite covers of normal spaces.
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.
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.