Particular Point Topology is Topology

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Theorem

Let $T = \left({S, \tau_p}\right)$ be a particular point space.

Then $\tau_p$ is a topology on $S$, and $T$ is a topological space.


Proof

We have by definition that $\varnothing \in \tau_p$, and as $p \in S$ we have that $S \in \tau_p$.


Now let $U_1, U_2 \in \tau_p$.

By definition $p \in U_1$ and $p \in U_2$, and so $p \in U_1 \cap U_2$ by definition of set intersection.

So $U_1 \cap U_2 \in \tau_p$.


Now let $\mathcal U \subseteq \tau_p$.

We have that $\forall U \in \mathcal U: p \in U$.

Hence from Subset of Union $p \in \bigcup \mathcal U$.


So all the properties are fulfilled for $\tau_p$ to be a topology on $S$.

$\blacksquare$


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