Region Less One Point is Region

Theorem
Let $$M = \left({A, d}\right)$$ be a metric space.

Let $$R \subseteq M$$ be a region of $$M$$.

Let $$\zeta \in R$$.

Then $$R - \left\{{\zeta}\right\}$$ is also a region of $$M$$.

Proof
From the definition, a region is a non-empty, open, path-connected subset of $$M$$.


 * First, note that as $$R$$ is open it can not be a singleton from Finite Subspace of Metric Space is Not Open.

Therefore $$R - \left\{{\zeta}\right\}$$ is not empty.


 * Next, we see that from Open Set Less One Point is Open that $$R$$ is open.


 * Now, let $$\alpha, \beta \in R$$.

As $$R$$ is path-connected, we can join $$\alpha$$ and $$\beta$$ with a Definition:Path $$\Gamma$$.

If $$\zeta \notin \Gamma$$, then $$\Gamma$$ is also a path in $$R - \left\{{\zeta}\right\}$$, and we are done.

If $$\zeta \in \Gamma$$, then we consider the $\epsilon$-neighborhood $$N_\epsilon \left({\zeta}\right)$$ of $$\zeta$$ for some $$\epsilon$$ such that $$N_\epsilon \left({\zeta}\right) \subseteq R$$.