# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Warning 3.1.4.10. The converse of Proposition 3.1.4.9 is false. For example, let $f: X \rightarrow S$ be a continuous function between topological spaces where $S = \ast$ consists of a single point. In this case, the function $f$ is a covering map if and only if the topology on $X$ is discrete. However, the induced map of simplicial sets $\operatorname{Sing}_{\bullet }(f): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is a covering map if and only if the simplicial set $\operatorname{Sing}_{\bullet }(X)$ is discrete: that is, if and only if every continuous function $[0,1] \rightarrow X$ is constant (Example 3.1.4.13). Many non-discrete topological spaces satisfy this weaker condition (for example, we could take $X$ to be the Cantor set).