# Convex Set is Simply Connected

## Theorem

Let $\struct {V, \tau}$ be a topological vector space over $\R$ or $\C$.

Let $A \subseteq V$ be a non-empty convex set.

Let $\tau_A$ be the subspace topology on $A$ induced by $\tau$.

Then $\struct{ A, \tau_A }$ is simply connected.

## Proof

Follows from Convex Set is Star Convex Set and Star Convex Set is Simply Connected.

$\blacksquare$