Loose Topology and Leisurely Mountain Topology
Definition
$X$ is said to be an infinite set.
- $\mathscr{T}_{f} : = \left\{ \emptyset , X \right\} \cup \left\{ U \subset X : | X \setminus U | < \infty \right\}$ is called the cofinite topology.
- $\mathscr{T}_{c} : = \left\{ \emptyset , X \right\} \cup \left\{ U \subset X : | X \setminus U | = \aleph_{0} \right\}$ is called the cocountable topology.
- Aleph null $\aleph_{0}$ refers to the cardinality of infinite countable sets.
Explanation
Though the terminology might seem daunting, essentially, it speaks of a topology where the complement is either finite or countable.
The cofinite topology is meaningless if $X$ is not an infinite set, and the cocountable topology is irrelevant if $X$ is not an uncountable set. If that were the case, omitting any point would result in $X \setminus U$ being either finite or countable respectively, ultimately leading to a discrete space.
While unique and requiring a bit of twisted thinking, becoming familiar with these properties is not straightforward, and rather than possessing truly important properties, they are often learned to find counterexamples to certain propositions.
Let’s try proving the following properties ourselves to get acquainted with the concepts of cofinite and cocountable.
Theorem
For a subspace $A$ of a cofinite space $X$ and a subspace $B$ of a cocountable space $Y$, the following holds:
- [1]: If $A$ is an infinite set, then $A ' = X$
- [2]: If $A$ is a finite set, then $A ' = \emptyset$
- [3]: If $B$ is an uncountable set, then $B ' = Y$
- [4]: If $B$ is a countable set, then $B ' = \emptyset$