Question

Let X be a topological space and let C and U be subsets of X. Define C to be closed if C contains all its limit points and define U to be open if every point p ∈ U has a neighborhood which is contained in U. Assuming these definitions show that the following statements are equivalent for a subset S of X. Which of the following statements is equivalent to the statement "S is closed in X"?

(A) X – S is open in X.
(B) S = ¯S.
(C) S contains all its limit points.
(D) All of the above.

Answer 1

The statement "S is closed in X" is equivalent to the statement (A) X – S is open in X. To prove this, we need to show that if S is closed, then X – S is open, and if X – S is open, then S is closed.

Step-by-step explanation:

The statement "S is closed in X" is equivalent to the statement (A) X – S is open in X. To prove this, we need to show that if S is closed, then X – S is open, and if X – S is open, then S is closed.

If S is closed, it means that S contains all its limit points. This implies that every point in X – S has a neighborhood that is contained in X – S. Therefore, X – S is open.

On the other hand, if X – S is open, it means that every point in X – S has a neighborhood that is contained in X – S. This implies that X – (X – S) = S contains all its limit points. Therefore, S is closed.