Completely Regular Space

A topological space X is said to be a completely regular space if every closed set A in X and a point x \in X, x \notin A, then there exists a continuous function f:X \to \left[ {0,1} \right], such that f\left( x \right) = 0 and f\left( A \right) = \left\{ 1 \right\}.

In other words, a topological space X is said to be a completely regular space if for any x \in X and a closed set C not containing x, there exists a continuous function f:X \to \left[ {0,1} \right] such that f\left( x \right) = 0 and f\left( C \right) = 1.

Remark:

Let us consider a continuous function g:X \to \left[ {0,1} \right] defined as g\left( x \right) = 1 and g\left( A \right) = 0. Since the constant function is continuous therefore taking the function g\left( x \right) = I\left( x \right) - f\left( x \right), where I\left( x \right) = 1,{\text{ }}\forall x \in X.

Now

 g\left( x \right) = I\left( x \right) - f\left( x \right) = 1 - 0 = 1

And

 g\left( A \right) = I\left( A \right) - f\left( A \right) = 1 - 1 = 0

Moreover the continuous function defined in the condition for a completely regular space is said to separate point x from the set A.

 

Tychonoff Space

A completely regular {T_1} space is said to be a Tychonoff space or a {T_{3\frac{1}{2}}}-space.

Note: It may be noted that since the product of {T_1} space is a {T_1} 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 {T_1} space is a Hausdorff space or {T_2} 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.