On first countable and minimal topological spaces

الملخص

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 be a collection of a topological spaces and , then X is first countable and Hausdorff – closed if and only if each x(n) is first countable and Hausdorff – closed