# Completely Regular Space

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

**Remark:**

Let us consider a continuous function defined as and . Since the constant function is continuous therefore taking the function , where .

Now

And

Moreover the continuous function defined in the condition for a completely regular space is said to separate point from 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 the product of space is a space and the product of a completely regular space is a completely regular space, so the product of a Tychonoff space is a Tychonoff space.

**Theorems**

• Every completely regular space is a regular space as well.

• Every completely regular space is a Hausdorff space or space.

• Every subspace of a completely regular space is a completely regular space.

• The product of a completely regular space is a completely regular space.

• Every subspace of a Tychonoff space is a Tychonoff space.

• Every Tychonoff space is a Hausdorff space.