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

TheoremA 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 set

\[\emptyset\]

henceis 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.