Theorem
A path connected space is always connected. The converse is not true. That is,
is a connected space does not imply it is connected.
Proof
Let
and
be subsets of
is a closed interval andis a continuous image of a closed interval. Hence both are connected. Each point ofis an accumulation point of
henceand B are not separated.
Hence
is connected.
But
is not path connected. There is no path connecting any point ofwith any point ofsinceandhave no points in common.
Hence connected does not imply path connected.