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
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
An open subspace $U$ of a space $X$ is a subspace of $X$ whose elements are “stable to small perturbations”, or can be “observed to belong to $U$ by measurements with finite precision”.
Depending on what sort of “space” $X$ is, this may be defined in different ways.
If $X$ is a topological space in the classical sense, then it is defined by specifying a collection of subsets to be called “open” (that are closed under finite intersections and arbitrary joins, i.e. are a sub-frame of the power set of $X$).
If $X$ is a convergence space (or some variation such as a subsequential space), we define a subset $U\subseteq X$ to be open if any filter/net/sequence converging to a point of $U$ must be eventually in $U$. This defines an “underlying topological space” of $X$.
In locale theory, every open $U$ in the locale defines an “open sublocale” which is given by the open nucleus
The idea is that this subspace is the part of $X$ which involves only $U$, and we may identify $V$ with $U \Rightarrow V$ when we are looking only at $U$. If $X$ is a (sober) topological space regarded as a locale, then any such open sublocale is spatial and coincides with the subspace determined by the subset $U$ (and this is true even in constructive mathematics).
See open subtopos.
In synthetic topology, we interpret ‘space’ as meaning simply ‘set’ (or type, i.e. the basic objects of our foundational system). There are then multiple ways to define “open subset”.
One is to use a dominance, which specifies the open subsets representably by a set of “open truth values”.
Another possibility is the following definition due to Penon: $U\subseteq X$ is open if for any $x\in U$ and $y\in X$, either $x\neq y$ or $y\in U$. This definition does not require a choice of dominance, but it is generally only correct for Hausdorff spaces; for instance, the open point in the Sierpinski space is not open in this sense. However, in various gros toposes of topological, smooth, or algebraic spaces it does induce the correct notion of open subsets for Hausdorff spaces; see Dubuc-Penon.
A subspace $A$ of a space $X$ is open if the inclusion map $A \hookrightarrow X$ is an open map.
The interior of any subspace $A$ is the largest open subspace contained in $A$, that is the union of all open subspaces of $A$. The interior of $A$ is variously denoted $Int(A)$, $Int_X(A)$, $A^\circ$, $\overset{\circ}A$, etc.
Jacques Penon, Topologie et intuitionnisme
Eduardo Dubuc and Jacques Penon, Objets compactes dans les topos