The discussion revolves around properties of interior points and closures in the context of metric spaces. Participants are examining the relationship between the set of interior points of a subset and the closure of that subset.
The discussion includes attempts to prove specific theorems and clarify definitions. Some participants have made progress in one direction of a theorem, while others express uncertainty about proving the opposite direction. There is an ongoing exploration of definitions and properties related to closures and interior points.
Participants are working within the constraints of a metric space and discussing the implications of open sets and their intersections with arbitrary subsets. There is mention of using contradiction in proofs, and some participants are questioning the clarity of certain statements made in the discussion.
golriz said:Is it true?
" Set of interior points of the closure of A equals the set of interior points of A. "
golriz said:I want to prove this theorem:
" Let U be an open subset of a metric space X,and A be an arbitrary subset of X. Prove that the intersection of U and closure of A is empty if and only if the intersection of U and A be empty. "
I've proved one direction of this theorem:
If the intersection of U and closure of A is empty then the intersection of U and A is empty too.
The closure of A is equal to the union of A and the set of all limit points(accumulation points) of A. Then we can use this definition of the closure of A. then after substitution, we have:
[ intersection of A and U ] U [intersection of U and the set of limit points of A] = empty set
so it says that both the sets [ intersection of A and U ] and [intersection of U and the set of limit points of A] should be empty.