Definition:Kolmogorov Space/Definition 1

Definition

Let $T = \struct {S, \tau}$ be a topological space.

$\struct {S, \tau}$ is a Kolmogorov space or $T_0$ space if and only if:

$\forall x, y \in S$ such that $x \ne y$, either:

$\exists U \in \tau: x \in U, y \notin U$

or:

$\exists U \in \tau: y \in U, x \notin U$

That is:

for any two distinct elements $x, y \in S$ there exists an open set $U \in \tau$ which contains one of the elements, but not the other.

That is:

$\struct {S, \tau}$ is a $T_0$ space if and only if every two elements in $S$ are topologically distinguishable.

Also see

• Equivalence of Definitions of $T_0$ Space

• Results about $T_0$ spaces can be found here.

Source of Name

This entry was named for Andrey Nikolaevich Kolmogorov.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Kolmogorov space