@ alice8675309

Here is a way to think about both the interior of a set and the closure of a set.

The interior of a set is the ‘largest’ open subset of the set.

The closure of a set is the ‘smallest closed set containing the set as a subset.

Here is another suggestion. If you can find the small bookElementary Theory of Metric Spacesby Robert B Reisel, then that would be a great introduction for you. It is written as a self-study or Moore-style class.