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Exercise 1191Topology › Topology of ℝⁿL1Added 3 Nov 2026

Convex sets are connected

#connectedness#convexity#segments

Statement

Show that a convex set of Rn\mathbb{R}^n is connected.

Hint

Path-connected sets are connected; convex sets are path-connected.

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Solution

Path-connectedness. As shown, every pair of points of a convex set is joined by a segment inside the set.

Connectedness. Path-connected sets are connected.

Conclusion. Convex sets are connected.

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