A topological space X is said to be completely regular space, if every closed set A in X and a point , , then there exist a continuous function , such that  and .
            In other words, a topological space X is said to be a completely regular space if for any  and a closed set C not containing x, there exist a continuous function  such that  and .

Remark:
            Let us consider a continuous function  defined as  and . Since constant function is continuous therefore taking the function , where .
            Now
                               
            And
                               
            Moreover the continuous function defined in the condition for completely regular space is said to be separate the point x form the set A.

Tychonoff Space:
            A completely regular T1-space is said to be a Tychonoff space or a -space.

Note: It may be noted that since product of T1-space is T1-space and product of completely regular space is completely regular space, so product of Tychonoff space is Tychonoff space.

Theorems:

• Every completely regular space is a regular space as well.
• Every completely regular T1-space is Hausdorff space or T2-space.
• Every subspace of a completely regular space is completely regular space.
• Product of completely regular space is a completely regular space.
• Every subspace of Tychonoff space is Tychonoff space.
• Every Tychonoff space is Hausdorff space.