Point in Discrete Space is Neighborhood

From ProofWiki
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