All sets contain their interior

up:: Interior (Topology)

Let be a Topological Space, and .

Then, by the definition of Interior (Topology), we know that is the union of open sets which are contained in .

Since all of them are contained in , their union will also be inside . The “lower bound” of its interior is , since is closed; the “upper bound” is itself, since all open sets involved are ; thus, .

Ad absurdum, suppose .
Then , absurd.