Here’s a strange phenomenon that I ran into when writing a MathOverflow answer a few years ago.
Lemma. Let be a set endowed with the cofinite topology, and assume is path connected. Then is contractible.
The assumption is for example satisfied when , for then any injection is a path from to . Path connectedness of cofinite spaces is related to partitioning the interval into disjoint closed subsets; see the remark below for some bounds on the cardinalities.
Proof. The result is trivial if is finite, for then both are equivalent to . Thus we may assume that is infinite. Choose a path from some to some . This induces a continuous map . Choose a bijection
and extend to a map by and for all . Then is continuous: the preimage of is if , and if , both of which are closed. Thus is a homotopy from to the constant map , hence a contraction.
I would love to see an animation of this contraction as goes from to … I find especially the slightly more direct argument for given here elusive yet somehow strangely visual.
Remark. If is countable (still with the cofinite topology), then is path connected if and only if . In the finite case this is clear (because then is discrete), and in the infinite case this is a result of Sierpiński. See for example this MO answer of Timothy Gowers for an easy argument.
There’s also some study of path connectedness of cofinite topological spaces of cardinality strictly between and , if such cardinalities exist. See this MO question for some results. In particular, it is consistent with ZFC that the smallest cardinality for which is path connected is strictly smaller than .