I l Show that if X has the discrete topology then its only c

I l

Show that if X has the discrete topology, then its only connected subspaces are point sets.

Solution

Let X have discrete topology.

Any subset Y of X is the union of singletons , each of which is both open and closed (as X has discrete topology)

So the maximal sets which cannot expressed as a disjoint union of open (closed) subsets are the singletons.

Hence the connected components of X are the singletons.