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 1] - traditionally (as mentioned) [ilmath]v=1[/ilmath] is used.
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
