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
A topological space $X$ is (semi-)locally simply connected if every neighborhood of a point has a subneighbourhood in which loops based at the point in the subneighborhood can be contracted in $X$. It is similar to but weaker than the condition that every neighborhood of a point has a subneighborhood that is simply connected. This latter condition is called local simple-connectedness.
A topological space $X$ is semi-locally simply-connected if it has a basis of neighbourhoods $U$ such that the inclusion $\Pi_1(U) \to \Pi_1(X)$ of fundamental groupoids factors through the canonical functor $\Pi_1(U) \to codisc(U)$ to the codiscrete groupoid whose objects are the elements of $U$. The condition on $U$ is equivalent to the condition that the homomorphism $\pi_1(U, x) \to \pi_1(X, x)$ of fundamental groups induced by inclusion $U \subseteq X$ is trivial.
(circle is locally simply connected)
is locally simply connected
By definition of the subspace topology and the defining topological base of the Euclidean plane, a base for the topology of $S^1$ is given by the images of open intervals under the local homeomorphism
But these open intervals are simply connected this exampleroup#EuclideanSpaceFundamentalGroup).
Semi-local simple connectedness is the crucial condition needed to have a good theory of covering spaces, to the effect that the topos of permutation representations of the fundamental groupoid of $X$ is equivalent to the category of covering spaces of $X$.
This is the fundamental theorem of covering spaces, see there for more.
For a topos-theoretic notion of locally $n$-connected see locally n-connected (infinity,1)-topos.