see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
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
subsets are closed in a closed subspace precisely if they are closed in the ambient space
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
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
Basic homotopy theory
There are many situations in algebraic topology in which one wants to work in a “Really Big Space”. Often, it is not that important which space is used, so long as it has some basic properties. A common feature that one wants of these spaces is that certain derived spaces be contractible. In many cases, it is possible to write down an actual contraction (rather than arguing from general nonsense) and in those cases, the contraction often uses the ability to “shift” pieces of the space from one place to another. This leads us to consider the notion of a shift map, and its cousin a split map, which provide enough structure to define the required contractions.
In short, a shift map is a generalisation of the obvious shift map $\ell^0 \to \ell^0$ given by $(x_1,x_2,x_3,\dots) \mapsto (0,x_1,x_2,x_3,\dots)$. It is an inclusion (even an embedding) and its eventual image, $\bigcap_k \im S^k$, is zero. Note that a shift map on $V$ induces an isomorphism $V \cong \mathbb{R} \oplus V$, but the existence of a shift map is a stronger condition than that.
A split map is similar, except that the induced decomposition is $V \cong V \oplus V$.
Let $V$ be a locally convex topological vector space over $\mathbb{R}$. Let $k \in \mathbb{N}$. A shift map of order $k$ on $V$ is a continuous linear map $S \colon V \to V$ with the following properties:
A locally convex topological vector space that admits a shift map will be called a shiftable space. The pair $(V,S)$ will be called a shift space.
Let $V$ be a locally convex topological vector space over $\mathbb{R}$. A split map on $V$ is a continuous linear map $S \colon V \to V$ with the following properties:
A locally convex topological vector space that admits a split map will be called a splittable space. The pair $(V,S)$ will be called a split space.
There are obvious generalisations for other fields than $\mathbb{R}$.
This is new terminology, invented to give a consistent way to refer to these spaces with their properties. At time of writing no existing name for this was known.