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Wikipedia https://en.wikipedia.org/wiki/Connected_space#Path_connectedness here claims a finite topological space is connected if and only if it is path-connected. Perhaps this is because connectedness and local path-connectedness imply path-connectedness and finite topological spaces are locally path-connected.

an indiscrete topological space, say $\left\{0,1\right\}$ with the topology $\tau=\left\{\emptyset,\left\{0,1\right\}\right\}$ is path-connected, so there must be a continuous map $f:I\to\left\{0,1\right\}$.

A map $f:X\to Y$ between two topological spaces is continuous if the pre-image of each open set in $Y$ is open in $X$, but what if the path is not a surjective map?