Definition:Kolmogorov Space/Definition 1

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

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

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

Source of Name

This entry was named for Andrey Nikolaevich Kolmogorov.