Proof That a Space With the Discrete Topology is Totally Disconnected and Locally Connected
A space
![](/university-maths/topology/proof-that-a-space-with-the-discrete-topology-is-totally-disconnected-and-locally-connected_html_m32407604.gif)
Proof
The only connected subsets of a discrete space are the singleton sets
![](/university-maths/topology/proof-that-a-space-with-the-discrete-topology-is-totally-disconnected-and-locally-connected_html_m3566169f.gif)
\[\emptyset\]
hence![](/university-maths/topology/proof-that-a-space-with-the-discrete-topology-is-totally-disconnected-and-locally-connected_html_m75009d54.gif)
A space
![](/university-maths/topology/proof-that-a-space-with-the-discrete-topology-is-totally-disconnected-and-locally-connected_html_m32407604.gif)
![](/university-maths/topology/proof-that-a-space-with-the-discrete-topology-is-totally-disconnected-and-locally-connected_html_41c3f12.gif)
![](/university-maths/topology/proof-that-a-space-with-the-discrete-topology-is-totally-disconnected-and-locally-connected_html_7003534.gif)
![](/university-maths/topology/proof-that-a-space-with-the-discrete-topology-is-totally-disconnected-and-locally-connected_html_m442ccad4.gif)
![](/university-maths/topology/proof-that-a-space-with-the-discrete-topology-is-totally-disconnected-and-locally-connected_html_60c2aa05.gif)
![](/university-maths/topology/proof-that-a-space-with-the-discrete-topology-is-totally-disconnected-and-locally-connected_html_1ae98962.gif)
![](/university-maths/topology/proof-that-a-space-with-the-discrete-topology-is-totally-disconnected-and-locally-connected_html_47e117c6.gif)
Letbe a space with the discrete topology. If
then the open sets consists of any selection of these. Take
and
be any selection from
including
then
and
and
is locally connected.