Exercise 1190Topology › Topology of ℝⁿL1Added 3 Nov 2026

Connectedness in $\mathbb{R}^n$

#connectedness#path-connectedness

Statement

Define connectedness and path-connectedness of subsets of $\mathbb{R}^n$ , and show that any convex set is path-connected.

Hint

For convex $C$ , the segment between any two points lies in $C$ .

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Solution

Definitions.
- $A$ is connected if no separation $A=U\cup V$ with $U,V$ nonempty open, disjoint.
- $A$ is path-connected if any two points are joined by a continuous path in $A$ .

Convex $\Rightarrow$ path-connected. For $x,y\in C$ convex, the segment $t\mapsto(1-t)x+ty$ , $t\in[0,1]$ , lies in $C$ .

Conclusion. Convex sets are path-connected (hence connected).

Connectedness in $\mathbb{R}^n$

Statement

Solution

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