On first countable and minimal topological spaces

الملخص EN

In this paper, we study the concept of minimal topological spaces and its relation with first countable space, we prove that if X is first countable completely regular space, then the following are equivalent.

• X is first countable and minimal completely regular space.

• X is first countable and completely regular – closed space.

A first countable and minimal Urysohn is semi regular and Let} :)({Mnnx? be a collection of a topological spaces and )(nxX? =, then X is first countable and Hausdorff – closed if and only if each x (n) is first countable and Hausdorff – closed.