Existence of Connected Punctiform Space
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Theorem
There exists at least one example of a connected topological space which is also punctiform.
Proof
Let $T$ be Cantor's leaky tent.
From Cantor's Leaky Tent is Connected, $T$ is a connected topological space.
From Cantor's Leaky Tent is Punctiform, $T$ is a punctiform space.
Hence the result.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness: Biconnectedness and Continua