Arens-Fort Topology is Topology
Theorem
Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $\tau$ is a topology on $T$.
Proof
We have that $\tuple {0, 0} \notin \O$ so $\O \in \tau$.
We have that $\forall m: \set {n: \tuple {m, n} \notin S} = \O$ which is finite, so $S \in \tau$.
Now consider $A, B \in \tau$, and let $H = A \cap B$.
If $\tuple {0, 0} \notin A$ or $\tuple {0, 0} \notin B$ then $\tuple {0, 0} \notin A \cap B$ and so $H \in \tau_p$.
Now suppose $\tuple {0, 0} \in A$ and $\tuple {0, 0} \in B$.
Then:
\(\ds H\) | \(=\) | \(\ds A \cap B\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \relcomp S H\) | \(=\) | \(\ds \relcomp S {A \cap B}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \relcomp S A \cup \relcomp S B\) | De Morgan's Laws: Complement of Intersection |
In order for $A$ and $B$ to be open sets it follows that:
- At most a finite number $m_A$ of sets $S_{m_A} = \set {n: \tuple {m_A, n} \notin A}$ are infinite
- At most a finite number $m_B$ of sets $S_{m_B} = \set {n: \tuple {m_B, n} \notin B}$ are infinite.
Let $r \in \Z_{\ge 0}$ and let:
- $S_{r_A} = \set {n: \tuple {r, n} \notin A}$
- $S_{r_B} = \set {n: \tuple {r, n} \notin B}$
We have that:
\(\ds \) | \(\) | \(\ds S_{r_{A \mathop \cap B} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {n: \tuple {r, n} \notin A \cap B}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {n: \tuple {r, n} \notin A} \cup \set {n: \tuple {r, n} \notin B}\) | De Morgan's Laws: Difference with Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds S_{r_A} \cup S_{r_B}\) |
If $S_{r_A}$ and $S_{r_B}$ are finite then so is $S_{r_A} \cup S_{r_B}$.
If only a finite number of $S_{m_A}$ and $S_{m_B}$ are infinite then it follows that only a finite number of $S_{m_{A \cap B}}$ are infinite.
So $A \cap B \in \tau$.
Now let $\UU \subseteq \tau$.
Then:
- $\ds \relcomp S {\bigcup \UU} = \bigcap_{U \mathop \in \UU} \relcomp S U$
by De Morgan's Laws: Complement of Union.
We have either of two options:
- $(1): \quad \forall U \in \UU: \tuple {0, 0} \notin U$
in which case:
- $\ds \tuple {0, 0} \notin \bigcap \UU$
Or:
- $(2): \quad$ At most a finite number $m$ of sets $S_m = \set {n: \tuple {m, n} \notin U}$ are infinite
in which case:
- At most a finite number $m$ of sets $S_m = \set {n: \tuple {m, n} \notin \bigcap \UU}$ are infinite.
So in either case $\ds \bigcup \UU \in \tau$.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $26$. Arens-Fort Space