13 - Path-Connectedness of a Connected Open Subset of a Normed Space

Path-Connectedness of a Connected Open Subset of a Normed Space

A connected open subset of a normed space is path-connected

Proof

Let $X$ be the connected open set

Suppose $P$ is a path-component of $X$
Let $a\in P$

Since $X$ is open then there exists ball $B(a,ϵ)$ contained in $X$
Let $b\in B(a,ϵ)$

So define explicit path $\gamma$ between $a$ and $b$ by

$$\gamma (t)=(1-t)a+tb$$

This is continuous and the image is contained in $B(a,ϵ)$ since

$$||\gamma (t)-a||=t||a-b||\le ||a-b||=d(a,b)<ϵ$$

for all $t\in [0,1]$

Hence $b\in P$ as it lies in the same path-component
So we have the path-components partition $X$ hence if there is more than one then
$X$ can be written as a disjoint union of non-empty open sets contrary to assumption

Hence there is only one path-component which is $X$ so $X$ is path-connected