A path-connected space is connected .
Proof
Let \( X \) be path-connected and fix \( p \in X \).
For every \( q \in X \), we denote the path image of the path
between \( p \) and \( q \) by \( I_q \). By the main theorem of connectedness
and (0x681d8519)
, \( B_q \) is connected.
Since \( X = \bigcup_{q \in X} B_q \) , (0x681ad889)
implies that \( X \) is connected.
Remarks
- If the space is locally path-connected , connectedness and path-connectedness are equivalent.