Cocountable Topology

Last updated Nov 1, 2022

Let $X$ be a Set. Then the Open sets $$\tau := {U \subset X : U^{C} \text{ is countable}} \cup {X, \emptyset}$$ make $X$ a Topological Space. This topology is known as the Cocountable Topology.

We simply need to check the Axioms defining a Topological Space.

1. By construction, ${X, \emptyset} \subset \tau$. $\checkmark$
2. Suppose $U, V \in \tau$. Then $$(U \cap V)^{C} = U^{C} \cup V^{C}$$ is Countable since A Union of Countable Sets is Countable. Thus $U \cap V \in \tau$.
3. Suppose ${U_\alpha}{\alpha \in A} \subset \tau$. Then $$(\bigcup\limits{\alpha \in A} U_{\alpha})^{C} = \bigcap\limits_{\alpha \in A} U_{\alpha}^{C} \subset U_{\alpha_{0}}^{C}$$ is Countable because $U_{\alpha_{0}}^{C}$ is Countable for some (really any) $\alpha_{0} \in A$ and Monotonicity of Cardinality. $\blacksquare$