Convex Set is Simply Connected
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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.
$\blacksquare$