# Fort Space is Zero Dimensional

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## Theorem

Let $T = \left({S, \tau_p}\right)$ be a Fort space on an infinite set $S$.

Then $T$ is a zero dimensional space.

## Proof

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

Then from Clopen Points in Fort Space, $\left\{{q}\right\}$ is clopen.

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

If we take the open neighborhoods of $p$ we get a local basis $\mathcal{U}_p$ with the following property:

- Since $p \in U \in \mathcal{U}_p$, 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.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 23 - 24: \ 9$