# Modified Fort Topology is Topology

## Theorem

Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.

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

## Proof

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

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

We have that $a, b \in S, S \setminus S = \O$ and $\O$ 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 $A \cap \set {a, b} \ne \O$ and $B \cap \set {a, b} \ne \O$.

Then:

 $\displaystyle H$ $=$ $\displaystyle A \cap B$ $\displaystyle \leadsto \ \$ $\displaystyle N \setminus H$ $=$ $\displaystyle N \setminus \paren {A \cap B}$ $\displaystyle$ $=$ $\displaystyle \paren {N \setminus A} \cup \paren {N \setminus B}$ 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 $N \setminus \paren {A \cap B}$ is finite.

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

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

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

$\displaystyle N \setminus \paren {\bigcup \UU} = \bigcap_{U \mathop \in \UU} \paren {N \setminus U}$

We have either of two options:

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

in which case:

$\displaystyle \bigcup \UU \subseteq N$

Or:

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

in which case:

$\displaystyle \bigcap_{U \mathop \in \UU} \paren {N \setminus U}$ is finite, from Intersection is Subset.

So in either case:

$\displaystyle \bigcup \UU \in \tau_{a, b}$

$\blacksquare$