Finite Complement Topology is Topology
Theorem
Let $T = \struct {S, \tau}$ be a finite complement space.
Then $\tau$ is a topology on $T$.
Proof
By definition, we have that $\O \in \tau$.
We also have that $S \in \tau$ as $\relcomp S S = \O$ which is trivially finite.
$\Box$
Let $A, B \in \tau$.
Let $H = A \cap B$.
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 |
But as $A, B \in \tau$ it follows that $\relcomp S A$ and $\relcomp S B$ are both finite.
Hence their union is also finite.
Thus $\relcomp S H$ is finite.
So $H = A \cap B \in \tau$ as its complement is finite.
$\Box$
Let $\UU \subseteq \tau$.
From De Morgan's laws: Complement of Union:
- $\ds \relcomp S {\bigcup \UU} = \bigcap_{U \mathop \in \UU} \relcomp S U$
But as:
- $\forall U \in \UU: \relcomp S U \in \tau$
each of the $\relcomp S U$ is finite.
Hence so is their intersection.
So $\ds \relcomp S {\bigcup \UU}$ is finite.
So by definition:
- $\ds \bigcup \UU \in \tau$
So $\tau$ is a topology on $T$.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 2$: Topological Spaces: Example $6$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.1$: Topological Spaces: Example $3.1.7$
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $18 \text { - } 19$. Finite Complement Topology