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
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
locally compact and second-countable spaces are sigma-compact
Theorems
Basic homotopy theory
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The singular cohomology (also Betti cohomology) of a topological space $X$ is the cohomology in ∞Grpd of its fundamental ∞-groupoid $\Pi(X)$:
for $\mathcal{B}^n \mathbb{Z} \in \infty Grpd$ the Eilenberg-MacLane object with the group $\mathbb{Z}$ in degree $n$, the degree $n$-singular cohomology of $X$ is
With $\infty Grpd$ presented by the category sSet of simplicial sets, the fundamental $\infty$-groupoid $\Pi(X)$ is modeled by the Kan complex
the singular simplicial complex of $X$.
The object $\mathcal{B}^n \mathbb{Z}$ is usefully modeled by the simplicial set
which is
the underlying simplicial set under the forgetful functor
from abelian simplicial groups to simplicial sets;
of the abelian simplicial group $\Xi \mathbb{Z}[n]$ which is the image under the Dold-Kan correspondence
of the chain complex
concentrated in degree $n$.
So in this model we have
A cocycle in this cohomology theory is a cochain on a simplicial set, on the singular complex $Sing X$.
Using the adjunction $(F \dashv U)$ this is isomorphic to
where
is the free abelian simplicial group on the simplicial set $Sing X$: this is the simplicial abelian group of singular chains of $X$. Its elements are formal sums of continuous maps $\Delta^n_{Top} \to X$. In this form
Using next the Dold-Kan adjunction this is
where
is the Moore complex of normalized chains of $\mathbb{Z}[Sing X]$: this is the complex of singular chains, formal sums over $\mathbb{Z}$ of simplices in $X$.
This way singular cohomology is the abelian dual of singular homology.
…
If the topological space $X$ is semi-locally contractible (meaning: any open subset $U\subset X$ has an open cover $W$ by open subsets $W_i\subset U$ that are contractible in $U$), then the sheaf cohomology of $X$ is isomorphic to the singular cohomology of $X$ for any abelian group of coefficients.
This was proved in (Sella 16).
A previous version of this entry led to the following discussion, which later led to extensive discussion by email. Partly as a result of this and similar discussions, there is now more information on how Kan complexes are $\infty$-groupoids at