Open Ball is Simply Connected

Theorem

Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space over $\R$ or $\C$.

Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the norm $\norm {\,\cdot\,}$ on $V$.

Let $\tau$ be the the topology on $V$ induced by the metric $d$.

Let $v \in V$ and $\epsilon \in \R_{>0}$.

Let $\map {B_\epsilon} v$ be the open $\epsilon$-ball of $v$.

Let $\tau_B$ be the subspace topology on $\map {B_\epsilon} v$ induced by $\tau$.

Then $\struct{ \map {B_\epsilon} v, \tau_B }$ is simply connected.

Proof

Normed Vector Space is Hausdorff Topological Vector Space shows that $\struct {V, \tau}$ is a topological vector space.

The result now follows from Open Ball is Convex Set and Convex Set is Simply Connected.

$\blacksquare$