Difference between revisions of "Discrete metric and topology"

Revision as of 23:34, 23 July 2015

Metric space definition

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

Open balls

The open balls of $X$ with the discrete topology are entirely $X$ or a single point, that is:

Open sets

The open sets of $(X,d_\text{discrete})$ consist of every subset of $X$ (the power set of $X$) - this is how the topology induced by the metric may be denoted $(X,\mathcal{P}(X))$

Discrete topology

The discrete topology on $X$ is the topology that considers every subset to be open. We may write $X$ imbued with the discrete topology as:

• $(X,\mathcal{P}(X))$ where $\mathcal{P}$ denotes power set

TODO: find reference - even though it is obvious as I show above that every subset is open

Related theorems

• Every map from a space with the discrete topology is continuous

Notes

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

References

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