CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A nice simplicial topological space is a simplicial topological space that satisfies certain extra properties that make it well behaved in homotopy theory, notably so that its geometric realization of simplicial spaces is its homotopy colimit.
Let $X : \Delta^{op} \to Top$ be a simplicial topological space.
Such $X$ is called
good if all the degeneracy maps $X_{n-1} \hookrightarrow X_n$ are all closed cofibrations;
proper if the inclusion $s X_n \hookrightarrow X_n$ of the degenerate simplices is a closed cofibration, where $s X_n = \bigcup_i s_i(X_{n-1})$.
In other words this says: $X_\bullet$ is proper if it is cofibrant in the Reedy model structure $[\Delta^{op}, Top_{Strom}]_{Reedy}$ on simplicial objects with respect to the Strøm model structure on Top.
The notion of good simplicial topological space goes back to (Segal), that of proper simplicial topological space to (May).
A good simplicial topological space is proper.
A proof appears as Lewis, corollary 2.4 (b). A generalization of this result is in RobertsStevenson.
For $X_\bullet$ any simplicial topological space, then ${|Sing X_\bullet|}$ is good, hence proper, and the natural morphism
is degreewise a weak homotopy equivalence.
This follows by results in (Lewis).
Since for $X \in Top$ the map $|Sing X| \to X$ is a cofibrant resolution in the standard model structure on topological spaces, we have that
is a degreewise weak homotopy equivalence. In particular each space $|Sing X_n|$ is a CW-complex, hence in particular a locally equi-connected space. By (Lewis, p. 153) inclusions of retracts of locally equi-connected spaces are closed cofibrations, and since degeneracy maps are retracts, this means that the degeneracy maps in $|Sing X_\bullet|$ are closed cofibrations.
Let $X_\bullet$ be a simplicial topological space. Then there is a natural weak homotopy equivalence
from its fat geometric realization of simplicial topological spaces to the homotopy colimit over the simplicial diagram $X : \Delta^{op} \to Top$.
If moreover $X_\bullet$ is proper, then the natural morphism ${\Vert X\Vert} \to {|X|}$ is a weak homotopy equivalence, and hence also the ordinary geometric realization is a model for the homotopy colimit.
That the geometric realization of simplicial topological spaces of a proper simplicial space is is homotopy colimit follows from the above fact that proper spaces are Reedy cofibrant, and using the general statement discussed at homotopy colimit about description of homotopy colimits by coends.
In the case $X_\bullet$ that is a good simplicial topological space, a direct (i.e., not using the fact that goodness implies properness) proof that $\Vert X\Vert \to |X|$ is a weak homotopy equivalence has been sketched by Graeme Segal and then refined by Tammo tom Dieck.
simplicial topological space, nice simplicial topological space
The definition of proper simplicial space goes back to
May originally said strictly proper for what now is just called proper .
The definition of good simplicial space goes back to
The implication $good \Rightarrow proper$ seems to be handled like a folk theorem. Its origin is maybe in
Comments on the relation between properness and cofibrancy in the Reedy model structure on $[\Delta^{op}, Set]$ are made in