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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]

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

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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