Proof That a Space With the Discrete Topology is Totally Disconnected and Locally Connected

Theorem

A spacewith the discrete topology is totally disconnected and locally connected.

|Proof

The only connected subsets of a discrete space are the singleton setsand the empty sethenceis totally disconnected.

A spaceis locally connected if forand any neighbourhoodofthere is a connected neighbourhoodofsuch that

Letbe a space with the discrete topology. Ifthen the open sets consists of any selection of these. Takeandbe any selection fromincluding thenandandis locally connected.