# Discrete metric and topology/Metric space definition

From Maths

Let [ilmath]X[/ilmath] be a set. The *discrete*^{[1]} metric, or *trivial metric*^{[2]} is the metric defined as follows:

- [math]d:X\times X\rightarrow \mathbb{R}_{\ge 0} [/math] with [math]d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\1 & \text{otherwise}\end{array}\right.[/math]

However any strictly positive value will do for the [ilmath]x\ne y[/ilmath] case. For example we could define [ilmath]d[/ilmath] as:

- [math]d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\v & \text{otherwise}\end{array}\right.[/math]
- Where [ilmath]v[/ilmath] is some arbitrary member of [ilmath]\mathbb{R}_{> 0} [/ilmath]
^{[Note 1]}- traditionally (as mentioned) [ilmath]v=1[/ilmath] is used.

- Where [ilmath]v[/ilmath] is some arbitrary member of [ilmath]\mathbb{R}_{> 0} [/ilmath]

**Note: however in proofs we shall always use the case [ilmath]v=1[/ilmath] for simplicity**

## Notes

- ↑ Note the
*strictly greater than 0*requirement for [ilmath]v[/ilmath]

## References

- ↑ Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene
- ↑ Functional Analysis - George Bachman and Lawrence Narici