# Completely Regular Space

A topological space is said to be completely regular space, if every closed set in and a point , , then there exist a continuous function , such that and .

In other words, a topological space is said to be a completely regular space if for any and a closed set not containing , 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 form the set .

**Tychonoff Space:**

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

**Note:** It may be noted that since product of space is 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 space is Hausdorff space or 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.