CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A filtered topological space $X_*$ is a filtered object in Top, hence
a topological space $X=X_\infty$
equipped with a sequence of subspaces
A filtered space $X_*$ is called a connected filtered space if it satisfies the following condition:
$(\phi)_0$: The function $\pi_0X_0 \to \pi_0 X_r$ induced by inclusion is surjective for all $r \geq 0$; and, for all $i \geq 1$, $(\phi_i): \pi_i(X_r,X_i,v)=0$ for all $r \gt i$ and $v \in X_0$.
There are two other forms of this condition which are useful under different circumstances.
A CW-complex $X$ with its filtration by skeleta $X^n$.
The free topological monoid $F X$ on a space $X$ filtered by the length of words. Given a based space $(X,x)$, there is also a reduced version by taking $F X$ and identifying $x$ with the identity of $F X$. This latter filtered space is known as the James construction $J(X,x)$, after Ioan James.
The James construction $J(X,x)$ can be constructed homotopy-theoretically, following ideas of Brunerie. Recall that, for $K$ a finite simplicial complex, for $(X,A)$ a pair of spaces, its polyhedral product $(X,A)^K$ is defined as the union $\bigcup_{\sigma\in S(K)}(X,A)^\sigma$ as a subspace of the Cartesian product $X^{V(K)}$. Here, for $\sigma\in S(K)$ a simplex of $K$, the subspace $(X,A)^\sigma$ consist of those $x\in X^{V(K)}$ such that, for each vertex $v$ in the complement of $\sigma$, the coordinate projection $\proj_v x$ lies in $A$. Equivalently, the polyhedral product $(X,A)^K$ can be considered as a homotopy colimit of these $(X,A)^\sigma$ over the poset $S(K)$ of simplexes $\sigma$ of $K$, where the maps are the respective inclusions.
For $X$ a space equipped with a basepoint $x$, define a filtered space $fil_\bullet$ as follows. Set $\fil_0$ as $\{x\}$. For $k\ge 1$, require that the following square is homotopy pushout:
where the unlabeled arrow is the homotopy colimit of a morphism of diagrams over $S(\partial \Delta[k-1])$ given by the maps
for each simplex $\sigma$ of the boundary simplicial complex $\partial \Delta[k-1]$ of the standard $(k-1)$-simplex.
For $(X,x)$ a pointed space, if $(X,x)$ is path-connected, then $fil_\infty \simeq \Omega\Sigma X$.
A similar example to the last using free groups instead of free monoids.
A similar example to the last using free groupoids on topological graphs.
A similar example to the last using the universal topological groupoid $U_\sigma(G)$ induced from a topological groupoid $G$ by a continuous function $f: Ob(G) \to Y$ to a space $Y$.
Examples of connected filtered spaces are:
The skeletal filtration of a CW-complex.
The word length filtration of the James construction for a space with base point such that $\{x\} \to X$ is a closed cofibration.
The filtration $(B C)_*$ of the classifying space of a crossed complex, filtered using skeleta of $C$.
This condition occurs in the higher homotopy van Kampen theorem for crossed complexes.
We thus see that filtered spaces arise from many geometric and algebraic situations, and see also stratified spaces). It is therefore interesting that one can define strict higher homotopy groupoids for filtered spaces more easily than for spaces themselves.
Note also that it is standard to be able to replace, using mapping cylinders, a sequence of maps $Y_n \to Y_{n+1}$ by a sequence of inclusions.
Guillaume Brunerie gave a talk entitled “The James Construction and $\pi_4(S^3)$” at the Institute of Advanced Studies on March 27, 2013. In this talk, he described James construction in homotopy type theory.