Fort Space is Zero Dimensional

From ProofWiki
Jump to navigation Jump to search


Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$.

Then $T$ is a zero dimensional space.


Let $q \in S$ such that $q \ne p$.

Then from Clopen Points in Fort Space, $\set q$ is clopen.

So $\forall q \in S, q \ne p: \set {\set q}$ is a local basis for $q$.

If we take the open neighborhoods of $p$ we get a local basis $\UU_p$ with the following property:

Since $p \in U \in \UU_p$, its complement does not contain $p$ and so it is open.

This implies that $U$ is also closed.

The union of the local basis forms a basis for the topology.

This basis is formed with clopen sets.

So, by definition, $T$ is a zero dimensional space.