Finite Complement Space is Locally Connected
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Theorem
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$.
Then $T$ is a locally connected space.
Proof
We have that a Finite Complement Space is Irreducible.
The result follows from Irreducible Space is Locally Connected.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $18 \text { - } 19$. Finite Complement Topology: $9$