Background: It's a fun exercise to try to construct a connected space $T$ such that no two points in $T$ can be connected with a path.
My solution to the puzzle was to use an order topology on a totally ordered set where the cardinality of any interval is $2^{2^{\aleph_0}}$, thus no two points can be connected with a path because the cardinality of $[0,1]$ is not big enough.
My question: Does there exist a connected space $T$ with cardinality equal to $2^{\aleph_0}$ which is no-where path connected? Equivalently, does there exist a connected space $T$ such that $|T| = 2^{\aleph_0}$ and any continuous function $f:[0,1]\rightarrow T$ is constant?
Best Answer
Gustin's Sequence Space $X$ is connected, but totally pathwise disconnected. A definition can be found here and it is also example 125 in Counterexamples in Topology (found here) by Lynn Arthur Steen and J. Arthur Seebach. It is countable though, but can probably be used to construct a space with fitting cardinality. Maybe the compact-open-topology on $X^X$ will do it, but that will require a few lemmas on how connectedness and totally pathwise disconnectedness is inherited.