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Counterexample

Is this true?

Connected ⇒ path-connected

No — it is false.

The counterexample

Topologist's sine curve {(x, sin 1/x) : 0 < x ≤ 1} ∪ {0}×[−1,1]

Why it fails

Connected as the closure of a connected set; no path reaches the segment.

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Keep practicing

Property not inherited

A property assumed to pass to subobjects, quotients, or through a chain. It does not.

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connectednesstopology

Go deeper· Explained in full in Connectedness — tests, preservation and the sine curve· From Metric Spaces › Connectedness and path-connectedness

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