Main theorem of connectedness

Let and be topological spaces and be a continuous function. If is (path)connected, then so is . #m/thm/topology

Proof for plain connectedness

Without loss of generality, consider a surjection . If is disconnected, it can be partitioned into open and , wherefore can be partitioned into open and and is thus disconnected. Alternatively, if is disconnected then there exists non-constant continuous , wherefore is nonconstant and continuous. Thus, if is compact so is its continuous image .

Proof for path-connectedness

Given any two points there exists a continuous function such that and . Clearly, constitutes a continuous function , and therefore for any two points there exists a Continuous path such that and . Thus is path connected.

This is a remarkably rare instance of properties being inherited by images, usually properties are inherited by preïmages.

Corollaries

Connectedness and Path connectedness are topological properties.
The quotient of a connected space is connected.
Connected fibres and quotient implies connected space

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