topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
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
continuous metric space valued function on compact metric space is uniformly continuous
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
Analysis Theorems
(see also Chern-Weil theory, parameterized homotopy theory)
For $X$ a topological space, the category of covering spaces $Cov(X)$ is the category whose
objects are covering spaces $E \overset{p}{\to} X$ over $X$;
morphisms are homomorphisms of covering spaces, hence continuous functions $f \colon E_1 \longrightarrow E_2$ such that we have a commuting diagram
This is equivalently the full subcategory of the slice category $Top_{/X}$ of the category of topological spaces over $X$ on those objects which are covering spaces.
(fundamental theorem of covering spaces)
For $X$ a topological space then forming monodromy is a functor from the category of covering spaces over $X$ to that of permutation groupoid representations of the fundamental groupoid of $X$:
If $X$ is locally path connected and semi-locally simply connected, then this is an equivalence of categories.
See at fundamental theorem of covering spaces for details.