Talk:Countable Finite Complement Space is not Path-Connected
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Proof question resolved
From an Explain template:
- Major hole in proof: $F$ needn't be countably infinite, which is necessary. Compare when $f$ is constant. It surely will be a path.
$T$ is a Definition:Countable Finite Complement Topology, which by definition is the Definition:Finite Complement Topology on a Definition:Countably Infinite Set.
The image of $f$ is $S$, which from Mapping Induces Partition on Domain has a countably infinite domain. (The inverse of a mapping is one-to-many, so the cardinality of the domain of $f$ is at least as big as that of its image.)
So yes, $F$ is countably infinite. --prime mover (talk) 03:13, 1 February 2018 (EST)