Interior preserves subset ordering

up:: Interior (Topology)

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

Since , it is “smaller” than , and thus it is expected that the amount of open sets which it contains will be lower to/the same as the amount which contains.

We can see it as follows: let be the collection of open sets which are contained inside . Then we have that

Since we expect , we prove that

Thus, interiors preserve subset ordering.