# Point in Discrete Space is Neighborhood

Jump to navigation
Jump to search

## Theorem

Let $T = \left({S, \tau}\right)$ be a discrete topological space.

Let $x \in S$.

Then $\left\{{x}\right\}$ is a neighborhood of $x$ in $T$.

## Proof

By definition, a neighborhood $N_x$ of $x$ is any subset of $S$ containing an open set which itself contains $x$.

That is:

- $\exists U \in \tau: x \in U \subseteq N_x \subseteq S$

From Set in Discrete Topology is Clopen we have that $\left\{{x}\right\}$ is open set in $S$.

So by Set is Subset of Itself, $\left\{{x}\right\}$ is a subset of $S$ containing an open set $\left\{{x}\right\}$ which itself contains $x$.

Hence the result.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 1 - 3: \ 7$