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Proof That a Connected Space is Not Necessarily Path Connected

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

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.

Henceis connected.

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.

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