Normed Vector Space is Locally Connected

Theorem

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.

Then $\struct {X, \norm {\, \cdot \,} }$ is locally connected.

Proof

Let $x \in X$.

Let $\epsilon > 0$.

Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball around $x$ in $\struct {X, \norm {\, \cdot \,} }$.

From Open Balls form Local Basis of Metric Space, the set $\BB_x = \set {\map {B_\epsilon} x : \epsilon > 0}$ is a local basis at $x$.

From Open Ball in Normed Vector Space is Connected, $\map {B_\epsilon} x$ is connected.

Hence $x$ has a local basis of connected sets.

Since $x$ was arbitrary, $\struct {X, \norm {\, \cdot \,} }$ is locally connected.

$\blacksquare$