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Counterexample

Is this true?

Closure of a path-connected set is path-connected

No — it is false.

The counterexample

Graph of sin(1/x) on (0,1]

Why it fails

Its closure is the topologist's sine curve.

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Property not inherited

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

See real exam options built on this mistakeDrill statements like this
connectedness

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

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