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
The Sierpiński space $\Sigma$ is the topological space
whose underlying set has two elements, say $\{0,1\}$,
whose set of open subsets is $\left\{ \emptyset, \{1\}, \{0,1\} \right\}$.
(We could exchange “0” and “1” here, the result would of course be homeomorphic).
Equivalently we may think of the underlying set as the set of of classical truth values $\{\bot, \top\}$, equipped with the specialization topology, in which $\{\bot\}$ is closed and $\{\top\}$ is an open but not conversely.
In constructive mathematics, it is important that $\{\top\}$ be open (and $\{\bot\}$ closed), rather than the other way around. Indeed, the general definition (since we can't assume that every element is either $\top$ or $\bot$) is that a subset $P$ of $\Sigma$ is open as long as it is upward closed: $p \Rightarrow q$ and $p \in P$ imply that $q \in P$. The ability to place a topology on $\Top(X,\Sigma)$ is fundamental to abstract Stone duality, a constructive approach to general topology.
This Sierpinski space
is a contractible space;
has a focal point;
is a sober topological space (but not T1).
According properties are inherited by the Sierpinski topos and the Sierpinski (∞,1)-topos over $Sierp$.
The Sierpinski space $S$ is a classifier for open subspaces of a topological space $X$ in that for any open subspace $A$ of $X$ there is a unique continuous function $\chi_A: X \to S$ such that $A = \chi_A^{-1}(\top)$.
Dually, it classifies closed subsets in that any closed subspace $A$ is $\chi_A^{-1}(\bot)$. Note that the closed subsets and open subsets of $X$ are related by a bijection through complementation; one gets a topology on the set of either by identifying the set with $\Top(X,\Sigma)$ for a suitable function space topology. (This part does not work as well in constructive mathematics.)
In synthetic topology, an analogue of the Sierpinski space is called a dominance.
Wikipedia, Sierpinski space
Paul Taylor, Foundations for computable topology – 7 The Sierpinski space (html)