Topological space

Discrete topology

On any set 𝑋 the discrete topology of 𝑋 is one where all subsets of 𝑋 are considered open (and therefore also closed, since the compliment of any subset is necessarily also a subset and therefore open), i.e. T =P(𝑋). Such a topology is the finest topology that can be formed on any set.

Metric

A discrete topology is metrizable with the so-called discrete metric

𝜌(𝑥1,𝑥2)={1if 𝑥≠𝑦0if 𝑥=𝑦

although there may exist other metricisations.

Properties

• A topological space is discrete iff every point is its own connected component

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