A strange contractible space

Here’s a strange phenomenon that I ran into when writing a MathOverflow answer a few years ago.

Lemma. Let X be a set endowed with the cofinite topology, and assume X is path connected. Then X is contractible.

The assumption is for example satisfied when |X| \geq |\mathbf R|, for then any injection f \colon [0,1] \hookrightarrow X is a path from x_0 = f(0) to x_1 = f(1). 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 X is finite, for then both are equivalent to |X| = 1. Thus we may assume that X is infinite. Choose a path f \colon [0,1] \to X from some x_0 = f(0) to some x_1 = f(1) \neq x_0. This induces a continuous map [0,1] \times X \to X \times X. Choose a bijection

    \[g \colon (X \setminus \{x_0,x_1\}) \times X \stackrel \sim\to X,\]

and extend to a map \bar g \colon X \times X \to X by g(x_0,x) = x and g(x_1,x) = x_1 for all x \in X. Then \bar g is continuous: the preimage of x \in X is g^{-1}(x) \cup (x_0,x) if x \neq x_1, and g^{-1}(x) \cup (x_0,x) \cup x \times X if x = x_1, both of which are closed. Thus \bar g \circ (f \times \mathbf 1_X) is a homotopy from \mathbf 1_X to the constant map x_1, hence a contraction. \qedsymbol

I would love to see an animation of this contraction as t goes from 0 to 1… I find especially the slightly more direct argument for |X| \geq |\mathbf R| given here elusive yet somehow strangely visual.

Remark. If X is countable (still with the cofinite topology), then X is path connected if and only if |X| = 1. In the finite case this is clear (because then X is discrete), and in the infinite case this is a result of SierpiƄski. 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 \aleph_0 = |\mathbf N| and \mathfrak c = |\mathbf R|, if such cardinalities exist. See this MO question for some results. In particular, it is consistent with ZFC that the smallest cardinality for which X is path connected is strictly smaller than \mathfrak c.

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