If $X$ is a connected Hausdorff space and $X$ has more than one point, then is $X$ infinite?
It’s well known that if $X$ is a connected metric space then the answer is in the affirmative. The proof is number 1 here:
https://www.math.stonybrook.edu/~claude/530mtm.pdf
So since every metric space is Hausdorff, the answer to the question is clearly yes. What would a non-metric proof look like?
If $X$ is a finite Hausdorff space then every singleton set is closed which implies that every subset is closed. Hence $X =\{x\} \cup (X \setminus \{x\})$ gives a disconnection of $X$ unless $X$ has only one point. So either $X$ has only one point or it is infinite.