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Math 2.0

General topology Example

Discrete metric

Example

Let $X$ be a set. For $x, y \in X$ , let $d(x, y) = 0$ if $x = y$ and $d(x, y) = 1$ if $x \neq y$ . Then $d$ is a metric on $X$ , the discrete metric.

Remarks

For each $x \in X$ , the open ball of radius $1/2$ about $x$ is $\{x\}$ . Thus each subset of $X$ is open, and the metric gives the discrete topology. A sequence converges in this metric if and only if it is eventually constant.

Further reading

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

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Tag	`0NA4FK7B8S`
URI	`m2://0NA4FK7B8S`
Kind	example
Writing	W0 stub
MSC	54E35
Instance of	Metric space

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