It is well known that if $X$ is a finite simplicial complex then for every continuous map $f:|X|\to |Y|$ there exists a simplicial map $F: X^{(n)}\to Y$ that $|F|$ is homotopic to $f$.
Does anyone know an example of a map $f$ where $X$ is infinite and $f$ cannot be homotopic with geometric realization of any map $F: X^{(n)}\to Y$ and any $n\in \mathbb N$.
Or maybe there are some generalizations of the simplicial approximation theorem or there are known theorems that say when a map $f:X\to Y$ can be approximated by a simplicial map $F:X^{(n)}\to Y$? For example for locally finite complexes?