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Is this true?

Path components are closed

No — it is false.

The counterexample

This one comes with the Notes pack.

See pricing

85 counterexamples are free — including most of the ones Part C leans on hardest.

The kind of mistake this is

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

Drill statements like this
connectednesstopology

Others that fail the same way

From Metric Spaces › Connectedness and path-connectedness

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