Proof That a Space With the Discrete Topology is Totally Disconnected and Locally Connected
A spacewith the discrete topology is totally disconnected and locally connected.
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.