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
A simplicial topological group is a simplicial object in the category of topological groups.
For various applications the ambient category Top of topological spaces is taken specifically to be
the category of compactly generated weakly Hausdorff spaces, or
or the category of k-spaces.
We take Top to be the category of k-spaces in the following.
A simplicial topological group $G$ is called well-pointed if for $*$ the trivial simplicial topological group and $i : * \to G$ the unique homomorphism, all components $i_n : * \to G_n$ are closed cofibrations.
For $B \in Top$ a fixed base object, it is often desirable to work in “$B$-parameterized families”, hence in the over-category $Top/B$ (see MaySigurdson). There is the relative Strøm model structure on $Top/B$.
A simplicial group in $G$ in $Top/B$ is called well-sectioned if for $B$ the trivial simplicial topological group over $B$ and $i : B \to G$ the unique homomorphism, all components $i_n : B \to G_n$ are $\bar f$-cofibrations.
Recall for a discrete simplicial group $G$ the notation $\bar W G \to W G$ for the Kan complex presentation of the universal principal infinity-bundle $\mathbf{E}G \to \mathbf{B}G$ from simplicial group. These constructions for discrete simplicial groups have immediate analogs for simplicial topological groups.
Let $G$ be a simplicial topological group. Write $\bar W G \in Top^{\Delta^{op}}$ for the simplicial topological space whose topological space of $n$-simplices is the product
in Top, equipped wwith the evident (…) face and degeneracy maps.
We say a morphism $f : X \to Y$ of simplicial topological spaces is a global Kan fibration if for all $n \in \mathbb{N}$ and $0 \leq k \leq n$ the canonical morphism
$\Lambda^n_k \in$ sSet $\hookrightarrow Top^{\Delta^{op}}$ is the $k$th $n$-horn regarded as a discrete simplicial topological space:
$sTop(-,-) : sTop^{op} \times sTop \to Top$ is the Top-hom object.
We say a simplicial topological space $X_\bullet \in Top^{\Delta^{op}}$ is (global) Kan simplicial space if the unique morphism $X_\bullet \to *$ is a global Kan fibration, hence if for all $n \in \mathbb{N}$ and all $0 \leq i \leq n$ the canonical continuous function
into the topological space of $k$th $n$-horns admits a section.
This global notion of Kan simplicial spaces is considered for instance in (BrownSzczarba) and (May).
Let $G$ be a simplicial topological group. Then
$G$ is a globally Kan simplicial topological space;
$\bar W G$ is a globally Kan simplicial topological space;
$W G \to \bar W G$ is a global Kan fibration.
The first statement appears as (BrownSzczarba, theorem 3.8), the second is noted in (RobertsStevenson), the third as (BrownSzczarba, lemma 6.7).
If $G$ is a well-pointed simplicial topological group, then
$G$ is a good simplicial topological space;
the geometric realization $|G|$ is well-pointed;
$\bar W G$ is a proper simplicial topological space.
The statement about $\bar W G$ is proven in (RobertsStevenson). The other statements are referenced there.
simplicial topological space, nice simplicial topological space
simplicial topological group
Basics theory of simplicial topological groups is in
and
Their principal ∞-bundles and geometric realization is discussed in
Discussion of homotopy theory over a base $B$ is in