Proof That a Space With the Discrete Topology is Totally Disconnected and Locally Connected
TheoremA space
with the discrete topology is totally disconnected and locally connected.Proof
The only connected subsets of a discrete space are the singleton sets
and the empty set \[\emptyset\]
hence
is totally disconnected.A space
is locally connected if for
and any neighbourhood
of
there is a connected neighbourhood
of
such that
Letbe a space with the discrete topology. If
then the open sets consists of any selection of these. Take
andbe any selection fromincluding
then
and
andis locally connected.