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
Recall the definition of a modified Fort space:

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 $H = A \cap B$.

Let either $A \subseteq N$ or $B \subseteq N$.

From Intersection is Subset we have that $A \cap B \subseteq N$ and $A \cap B \subseteq B$.

Hence $A \cap B \subseteq N$ from Subset Relation is Transitive.

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:

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:
 * $\ds N \setminus \paren {\bigcup \UU} = \bigcap_{U \mathop \in \UU} \paren {N \setminus U}$

We have either of two options:
 * $(1): \quad \forall U \in \UU: U \subseteq N$

in which case:
 * $\ds \bigcup \UU \subseteq N$

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

in which case:
 * $\ds \bigcap_{U \mathop \in \UU} \paren {N \setminus U}$ is finite, from Intersection is Subset.

So in either case:
 * $\ds \bigcup \UU \in \tau_{a, b}$