Theorem
If
is a path connected space then it is connected.
Proof
Letbe path connected and let
For each
there exists a path
from
to
Then
and
Since eachis connected - as a continuous image of the connected set
-
is connected.
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Theorem
Ifis a path connected space then it is connected.
Proof
Letbe path connected and let
For eachthere exists a pathfromto
Thenand
Since eachis connected - as a continuous image of the connected set-is connected.