Difference between revisions of "Discrete metric and topology/Metric space definition"

Revision as of 19:55, 23 July 2015

Let $X$ be a set. The discrete^[1] metric, or trivial metric^[2] is the metric defined as follows:

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

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

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

Note: however in proofs we shall always use the case $v=1$ for simplicity

Notes

1. Jump up ↑ Note the strictly greater than 0 requirement for $v$

References

1. Jump up ↑ Todo - this is how I was taught, can't find source
2. Jump up ↑ Functional Analysis - George Bachman and Lawrence Narici