Discrete metric and topology/Summary
From Maths
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:
