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 SierpiĆski. 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
.
Typo: You said “profinite” in your remark when you meant “cofinite.”
Thanks, fixed!