Given a space , there are many ways to define a topology. More common methods are one of the following:
- explicitly list out all the open sets
- define the rules for a subset to be open
- define the basis/subbasis of the topology (Note: The basis here refers to topological basis, this is different than a vector space basis that we have learnt in linear algebra.)
More often than not, we will work with topological spaces that has infinitely many open sets. In this case it will be hard to explicitly list out all the open sets in the topological space. In this post, we will look at a topology call cocountable topology, and luckily, we can describe the open set of this topology, thus we do not need to describe a topological basis.
Given a ground set , the cocountable topology on is the empty set together with the collection of subsets such that is countable, namely the complement of is countable. For example, if , then under the cocountable topology, none of the subsets of is open.
Now let’s assume that is an uncountable set. By definition, . Then, let be a collection of open sets in for some index set . Thus is countable for all . Now consider
by De Morgan’s law, where is arbitrary. Since is countable for all $\lambda \in \Lambda$ and intersections will only result in smaller set, therefore must be countable. Thus we may conclude that .
Finally, let be a finite collection. Then by De Morgan’s law, we have . Notice that finite union of countable sets is still countable (in fact, countable union of countable sets is countable), thus is countable and .
Therefore is indeed a topology.