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
On
A set $X$ is totally disconnected if the only connected subsets $S \subset X$ are point sets. Note that points are open in the discrete topology, which means that if $S \subset X$ contains two or more points, you can write $S$ as the union of its points, which are open and obviously disjoint $\rightarrow S$ will be disconnected. This means the only connected components will be points $\rightarrow X$ is totally disconnected.
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.