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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]

Its closure is the topologist's sine curve.

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
connectedness

Others that fail the same way

From Metric Spaces › Connectedness and path-connectedness

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