Sierpiński Space is Path-Connected

Theorem

Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.

Then $T$ is path-connected.


Proof

A Sierpiński space is a particular point space by definition.

A Particular Point Space is Path-Connected.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $11$. Sierpinski Space: $18$
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