Modified Fort Topology is Topology

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Theorem

Let $T = \left({S, \tau_{a, b}}\right)$ be a modified Fort space.


Then $\tau_{a, b}$ is a topology on $S$.


Proof

Let $S = N \cup \left\{{a, b}\right\}$ where $N$ is infinite, $a \ne b$ and $a, b \notin N$.


We have that $\varnothing \subseteq N$ so $\varnothing \in \tau_{a, b}$.


We have that $a, b \in S, S \setminus S = \varnothing$ and $\varnothing$ is trivially finite, so $S \in \tau_{a, b}$.


Now consider $A, B \in \tau_{a, b}$, and let $H = A \cap B$.


If $A \subseteq N$ and $B \subseteq N$ then $A \cap B \subseteq N$ from Intersection is Largest Subset.

So by definition $A \cap B \in \tau_{a, b}$


Now suppose $A \cap \left\{{a, b}\right\} \ne \varnothing$ and $B \cap \left\{{a, b}\right\} \ne \varnothing$.

Then:

\(\displaystyle H\) \(=\) \(\displaystyle A \cap B\)
\(\displaystyle \implies \ \ \) \(\displaystyle N \setminus H\) \(=\) \(\displaystyle N \setminus \left({A \cap B}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({N \setminus A}\right) \cup \left({N \setminus B}\right)\) De Morgan's Laws: Difference with Intersection


In order for $A$ and $B$ to be open sets we have that $N \setminus A$ and $N \setminus B$ are both finite.

Hence their union is also finite and so $N \setminus \left ({A \cap B}\right)$ is finite.

So $H = A \cap B \in \tau_{a, b}$ as its complement is finite.


Now let $\mathcal U \subseteq \tau_{a, b}$.

Then from De Morgan's Laws: Difference with Union:

$\displaystyle N \setminus \left({\bigcup \mathcal U}\right) = \bigcap_{U \mathop \in \mathcal U} \left({N \setminus U}\right)$


We have either of two options:

$(1): \quad \forall U \in \mathcal U: U \subseteq N$

in which case:

$\displaystyle \bigcup \mathcal U \subseteq N$


Or:

$(2): \quad \exists U \in \mathcal U: N \setminus U$ is finite

in which case:

$\displaystyle \bigcap_{U \mathop \in \mathcal U} \left({N \setminus U}\right)$ is finite, from Intersection is Subset.


So in either case:

$\displaystyle \bigcup \mathcal U \in \tau_{a, b}$

$\blacksquare$


Sources