Kolmogorov space ( $T_{0}$ )

A topological space is

T_{0}

when every pair of distinct points is distinguished by an open set containing exactly one of them.

Table of contents
- Sierpiński space Kolmogorov space
  - Embedding of a T0 space into a power of the Sierpiński space Sierpiński space

Sierpiński space ( $S$ )

The Sierpiński space is

{0, 1}

\emptyset

,

{1}

, and

{0, 1}

. Continuous maps

X \to S

are in bijection with open subsets of

X

through

g \mapsto g^{- 1} (1)

.

Embedding of a T0 space into a power of the Sierpiński space ( $X ↪ S^{Top (X, S)}$ )

X

, evaluation against all continuous maps

g : X \to S

is a homeomorphism onto its image under the map

x ⟼ (g (x))_{g}

into a power of the Sierpiński space. The

T_{0}

axiom makes it injective, and the coordinate inverse images of

{1}

recover every open subset of

X

.

 Ancestors (6)

 Incoming links (1)

Separation axiom

 Synonyms (1)

codex/t0-space

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (1)

Kolmogorov space by

Wikipedia Bot  1

A Kolmogorov space, also known as a \( T_0 \) space, is a type of topological space that satisfies a specific separation axiom. In a Kolmogorov space, for any two distinct points \( x \) and \( y \), there exists an open set containing one of the points but not the other. This means that for any two points in the space, it is possible to find an open set that "separates" them.

 Read the full article

  See all articles in the same topic Create my own version