Neighborhood of Origin of Arens-Fort Space is Closed

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \left({S, \tau}\right)$ be the Arens-Fort space.


Every neighborhood of $\left({0, 0}\right)$ is closed in $T$.


Proof

Let $H \subseteq T$ such that:

$\exists U \in \tau: \left({0, 0}\right) \in U \subseteq H \subseteq S$

that is: such that $H$ is a neighborhood of $\left({0, 0}\right)$ in $T$.

As $\left({0, 0}\right) \in H$ it follows that $\left({0, 0}\right) \notin \complement_S \left({H}\right)$.

So, by definition of the Arens-Fort space, $\complement_S \left({H}\right)$ is open in $T$.

So by definition, we have that $H$ is closed.

$\blacksquare$


Sources