In the page 10 of the paper "Filling Riemannian manifolds" by Gromov (ProjetEuclid link), the author proves the following inequality (1.2) relating the systole and the filling radius of manifolds. $$\operatorname{Sys}_1(M)\leq 6\operatorname{FillRad}(M)$$

During the proof, he uses an argument something like this:

*"$M$ is a closed $n$-dimensional manifold and $[M]$ is the fundamental class of it. In some ambient space $X$, the fundamental class $[M]$ is null-homologous. Therefore, one can find $n+1$-dimensional singular chain $c$ inside $X$ such that $\partial c=[M]$. Moreover, using a $\textbf{piecewise linear approximation}$ of $c$, one can construct a polyhedron $P$ such that $M\subset P \subset X$ and the fundamental class $[M]$ is null-homologous inside of this $P$."*

So, somehow he is constructing a triangulation of realization of the singular chain $c$ (i.e., union of continuous images of $\Delta_{n+1}$) using "piecewise linear approximation". But I don't understand this step. How do we guarantee this realization of singular chain is triangulable? It is not necessarily manifold. What is precise meaning of his "piecewise linear approximation" of singular chain?

*Edit:* I changed the title of this question from "When a topological space (not necessarily manifold) has a triangulation?" to "A question on the Gromov's proof of $\operatorname{Sys}_1(M)\leq 6\operatorname{FillRad}(M)$". Even though the original question is interesting in its own right, but I want to focus on one specific topic: Understanding Gromov's argument.

"When does a topological space (not necessarily manifold) have a triangulation?", it is an open question whether the (compact) space of closed subgroups of $\mathbf{R}^n$ has a triangulation for $n\ge 3$ (for $n\le 2$ it's true). $\endgroup$ – YCor Feb 9 at 0:35