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
Theorems
$Loc$ or $Locale$ is the category whose objects are locales and whose morphisms are continuous maps between locales. By definition, this means that $Loc$ is the opposite category of Frm, the category of frames.
$Loc$ is used as a substitute for Top if one wishes to do topology with locales instead of standard topological spaces.
$Loc$ is naturally a (1,2)-category; its 2-morphism are the pointwise ordering of frame homomorphisms.
The 2-category Locale has
as morphisms $f : X \to Y$ frame homomorphisms $f^* : Op(Y) \to Op(X)$;
a unique 2-morphisms $f \Rightarrow g$ whenever for all $U \in Op(Y)$ we have a (then necessarily unique) morphism $f^* U \to g^* U$.
(For instance Johnstone, C1.4, p. 514.)
For any base topos $S$ the 2-category $Loc(S)$ of internal locales in $S$ is equivalent to the subcategory of the slice of Topos over $S$ on the localic geometric morphisms. See there for more details.
See locale for more properties.
The 2-functor that forms categories of sheaves
exhibits $Locale$ as a full sub-2-category of Topos. See localic reflection for more on this.
For instance Section C1 of