Discrete metric and topology
Contents
Metric space definition
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
Metric summary
Property  Comment 

induced topology  discrete topology  which is the topology [ilmath](X,\mathcal{P}(X))[/ilmath] (where [ilmath]\mathcal{P} [/ilmath] denotes power set) 
Open ball  [ilmath]B_r(x):=\{p\in X\vert\ d(p,x)< r\}=\left\{\begin{array}{lr}\{x\} & \text{if }r\le 1 \\ X & \text{otherwise}\end{array}\right.[/ilmath] 
Open sets  Every subset of [ilmath]X[/ilmath] is open. Proof outline: as for a subset [ilmath]A\subseteq X[/ilmath] we can show [ilmath]\forall x\in A\exists r[B_r(x)\subseteq A][/ilmath] by choosing say, that is [ilmath]A[/ilmath] contains an open ball centred at each point in [ilmath]A[/ilmath]. 
Connected  The topology generated by [ilmath](X,d_\text{discrete})[/ilmath] is not connected if [ilmath]X[/ilmath] has more than one point. Proof outline:

Metric objects
Open balls
The open balls of [ilmath]X[/ilmath] with the discrete topology are entirely [ilmath]X[/ilmath] or a single point, that is:
 [math]B_r(x):=\{p\in X\vert\ d(x,p)<r\}=\left\{\begin{array}{lr}\{x\} & \text{for }r\le 1\\ X & \text{otherwise}\end{array}\right.[/math]
 By definition [math]B_r(x):=\{p\in X\vert\ d(x,p)<r\}[/math] note that for:
 [ilmath]r\le 1[/ilmath] we have [ilmath]B_r(x)=\{x\}[/ilmath] as
 [ilmath]d(x,p)< r \le 1[/ilmath] so [ilmath]d(x,p)<1[/ilmath] only when [ilmath]x=p[/ilmath], as if [ilmath]x\ne p[/ilmath] then [ilmath]d(x,p)=1\not<1[/ilmath] (proof by contrapositive)
 [ilmath]r> 1[/ilmath] we have [ilmath]B_r(X)=X[/ilmath] as
 [ilmath]d(x,y)\le 1[/ilmath] always, so we have [ilmath]\forall x,y\in X[d(x,y)\le 1< r][/ilmath] so [ilmath]\forall x,y\in X[d(x,y)< r][/ilmath] thus the ball contains every point in [ilmath]X[/ilmath]
 [ilmath]r\le 1[/ilmath] we have [ilmath]B_r(x)=\{x\}[/ilmath] as
This completes the proof
Open sets
The open sets of [ilmath](X,d_\text{discrete})[/ilmath] consist of every subset of [ilmath]X[/ilmath] (the power set of [ilmath]X[/ilmath])  this is how the topology induced by the metric may be denoted [ilmath](X,\mathcal{P}(X))[/ilmath]
Every subset of [ilmath]X[/ilmath] is an open set
 Let [ilmath]A[/ilmath] be a subset of [ilmath]X[/ilmath], we will show that [ilmath]\forall x\in A\exists r>0[B_r(x)\subseteq A][/ilmath]
 Let [ilmath]x\in A[/ilmath] be given
 Choose [ilmath]r=\tfrac{1}{2}[/ilmath]
 Now [ilmath]B_r(x)=\{x\}\subseteq A[/ilmath]
 This must be true as we know already that [ilmath]x\in A[/ilmath] (to show this formally use the impliessubset relation)
 Now [ilmath]B_r(x)=\{x\}\subseteq A[/ilmath]
 We have shown that given an [ilmath]x\in A[/ilmath] we can find an open ball about [ilmath]x[/ilmath] entirely contained within [ilmath]A[/ilmath]
 Choose [ilmath]r=\tfrac{1}{2}[/ilmath]
 We have shown that for any [ilmath]x\in A[/ilmath] we can find an open ball about [ilmath]x[/ilmath] entirely contained within [ilmath]A[/ilmath]
 Let [ilmath]x\in A[/ilmath] be given
This completes the proof
Discrete topology
The discrete topology on [ilmath]X[/ilmath] is the topology that considers every subset to be open. We may write [ilmath]X[/ilmath] imbued with the discrete topology as:
 [ilmath](X,\mathcal{P}(X))[/ilmath] where [ilmath]\mathcal{P} [/ilmath] 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
 Every continuous map from a nonempty connected space to a discrete space is constant
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