# 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

And

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.