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Math 2.0

General topology Standard examples of topologies Example

Discrete topology

Example

Let $X$ be a set and let $\tau = \mathcal{P}(X)$ be the set of all subsets of $X$ . Then $\tau$ is a topology on $X$ , the discrete topology. The space $(X, \tau)$ is a discrete space.

Remarks

Each subset of $X$ is both open and closed. Each point $x$ gives an open set $\{x\}$ . It is the finest topology on $X$ . It is the metric topology of the discrete metric. Each function from a discrete space to any topological space is continuous. A discrete space is compact if and only if it is finite.

Where this leads

2 pages build on this one.

Further reading

Web https://en.wikipedia.org/wiki/Discrete_space, Definitions

About this page

Tag	`0S2TEHX4E1`
URI	`m2://0S2TEHX4E1`
Aliases	`discrete-space`
Kind	example
Writing	W2 usable
MSC	54A05
Instance of	Topological space
Same as	Wikidata Q175116 Wikipedia Discrete space

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