Closure preserves subset ordering

up:: Closure (Topology)

Let be a Topological Space, and , and let be their closures.

Since , it is “smaller” than , and thus it is expected that the amount of closed sets which contain it will be higher to/the same as the amount which enclose .

We can see it as follows: let be the collection of closed sets which enclose . Then we have that

Since we expect , we prove that

Thus, closures preserve subset ordering.