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
(finer/coarser topologies)
Let $X$ be a set, and let $\tau_1, \tau_2 \in P(X)$ be two topologies on $X$, hence two choices of open subsets for $X$, making it a topological space. If
hence if every open subset of $X$ with respect to $\tau_1$ is also regarded as open by $\tau_2$, then one says that
the topology $\tau_2$ is finer than the topology $\tau_2$
the topology $\tau_1$ is coarser than the topology $\tau_2$.
(discrete and co-discrete topology)
Let $S$ be any set. Then there are always the following two extreme possibilities of equipping $X$ with a topology $\tau \subset P(X)$, and hence making it a topological space:
$\tau \coloneq P(S)$ the set of all open subsets;
this is called the discrete topology on $S$, it is the finest topology (def. 1) on $X$,
we write $Disc(S)$ for the resulting topological space;
$\tau \coloneqq \{ \emptyset, S \}$ the set contaning only the empty subset of $S$ and all of $S$ itself;
this is called the codiscrete topology on $S$, it is the coarsest topology (def. 1) on $X$
we write $CoDisc(S)$ for the resulting topological space.