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.
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.