A space is totally disconnected if its only connected subspaces are one-point
sets. Show that if $X$ has the discrete topology, then $X$ is totally disconnected.
Does the converse hold?
my attempts : i know that Totally disconnected means that no two points are in the same connected component. So for $x,y\in X$ there is no connected set containing both points. In particular the set $\{x,y\}$ is not connected, thus is discrete..
As i can not able to prove this
Pliz help me
Best Answer
As a counterexample, consider the set $$ X = \{1/n : n = 1,2,3,\dots\} \cup \{0\} \subset \Bbb R $$ Verify that $X$ is totally disconnected, but is not discrete since the subset $\{0\}$ fails to be open.