Just to make sure that I'm not overlooking anything, is the following an example of a quotient map [itex]p: X \to Y[/itex] with the properties that Y is pathwise connected (i.e. connected by a continuous function from the unit interval), [itex]\forall y \in Y: p^{-1}(\{ y \}) \subset X[/itex] also pathwise connected and such that X is(adsbygoogle = window.adsbygoogle || []).push({}); notconnected?

As [itex]X[/itex] take the topologist's sine curve (i.e. the closure of the curvy bit), and simply define p as the projection on the x-axis, such that [itex]Y = [0,1][/itex] with the quotient topology, which I think coincides with the normal topology on it. By definition of quotient topology, p is a quotient map. Also, Y is pathwise connected and it seems that [itex]\forall y \in [0,1]: p^{-1}(\{ y \})[/itex] is also pathwise connected.

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# Example in topology: quotient maps and arcwise connected

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