see also algebraic topology, functional analysis and homotopy theory
topological space (see also locale)
fiber space, space attachment
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
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
A space (such as a topological space) is second-countable if, in a certain sense, there is only a countable amount of information globally in its topology. (Change ‘globally’ to ‘locally’ to get a first-countable space.)
(second-countable topological space)
A topological space is second-countable if it has a base for its topology consisting of a countable set of subsets.
A locale is second-countable if there is a countable set $B$ of open subspaces (elements of the frame of opens) such that every open $G$ is a join of some subset of $B$. That is, we have
The weight of a space is the minimum of the cardinalities of the possible bases $B$. We are implicitly using the axiom of choice here, to suppose that this set of cardinalities (which really is a small set because bounded above by the number of open subspaces, and inhabited by this number as well) has a minimum. But without Choice, we can still consider this collection of cardinalities.
Then a second-countable space is simply one with a countable weight.
(Euclidean space is second-countable)
Let $n \in \mathbb{N}$. Consider the Euclidean space $\mathbb{R}^n$ with its Euclidean metric topology. Then $\mathbb{R}^n$ is second countable.
A countable set of base open subsets is given by the open balls $B^\circ_x(\epsilon)$ of rational radius $\epsilon \in \mathbb{Q}_{\geq 0} \subset \mathbb{R}_{\geq 0}$ and centered at points with rational coordinates: $x \in \mathbb{Q}^n \subset \mathbb{R}^n$.
A compact metric space is second-countable.
A separable metric space, e.g., a Polish space, is second-countable.
It is not true that separable first-countable spaces are second-countable; a counterexample is the real line equipped with the half-open or lower limit topology that has as basis the collection of half-open intervals $[a, b)$.
A Hausdorff locally Euclidean space is second-countable precisely it is paracompact and has a countable set of connected components. In this case it is called a topological manifold.
See at topological space this prop..
A countable coproduct (disjoint union space) of second-countable spaces is second-countable.
Countable products (product topological spaces) of second-countable spaces are second-countable.
Subspaces of second-countable spaces are second-countable.
If $X$ is second-countable and there is an open surjection $f \colon X \to Y$, then $Y$ is second-countable.
For second-countable T_3 spaces $X, Y$, if $X$ is locally compact, then the mapping space $Y^X$ with the compact-open topology is second-countable.
Cf. Urysohn metrization theorem and Polish space. I (Todd Trimble) am uncertain to what extent the $T_3$ assumption can be removed.