Proof That a Connected Space is Not Necessarily Path Connected
A path connected space is always connected. The converse is not true. That is,is a connected space does not imply it is connected.
Letandbe subsets of
is a closed interval andis a continuous image of a closed interval. Hence both are connected. Each point ofis an accumulation point ofhenceand B are not separated.
Butis 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.