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
For $S \subset X$ a subset of a topological space $X$, a, interior point of $S$ is a point $x \in S$ which has a neighbourhood in $X$ that is contained in $S$. The union of all interior points is the interior $S^\circ$ of $S$. It can be defined as the largest open set contained in $S$.
In general, we have $S^\circ \subseteq S$. $S$ is open if and only if $S^\circ = S$, so that in particular $S^{\circ\circ} = S^\circ$. This makes the interior operator $P(X) \to P(X): S \mapsto S^\circ$ a co-closure operator. It also satisfies the equations $(S \cap T)^\circ = S^\circ \cap T^\circ$ and $X^\circ = X$. Moreover, any co-closure operator $c$ on $P(X)$ that preserves finite intersections must be the interior operation for some topology, namely the family consisting of fixed points of $c$; this gives one of many equivalent ways to define a topological space.
Compare the topological closure $\bar{S}$ and frontier $\partial S = \bar{S} \setminus S^\circ$.
The interior of a subtopos $\mathcal{E}_j$ of a Grothendieck topos $\mathcal{E}$, as well as the exterior, were defined in an exercise in SGA4: $Int(\mathcal{E}_j)$ as the largest open subtopos contained in $\mathcal{E}_j$. The boundary of a subtopos is then naturally defined as the subtopos complementary to the (open) join of the exterior and interior subtoposes in the lattice of subtoposes.
Let $(X,\tau)$ be a topological space and let $S \subset X$ be a subset. Then the topological interior of $S$ equals the complement of the topological closure $Cl(X\backslash S)$ of the complement of $S$:
By taking complements once more, the statement is equivalent to
Now we compute: