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Example in topology: quotient maps and arcwise connected

  1. Jan 8, 2012 #1
    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 not connected?

    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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  3. Jan 8, 2012 #2

    micromass

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    Seems ok!! :smile:
     
  4. Jan 13, 2012 #3

    Bacle2

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    Isn't the topologists' sine curve connected? It is the closure of the continuous image of the semi-open interval (0,1]. And the closure of a connected space (which is the definition of the topologists' sine curve that I know of, i.e., the closure of the graph of Sin(1/x) for x in (0,1]) is also connected. AFAIK, the topologists' sine curve is the standard example of connected , but not path-connected.
     
    Last edited: Jan 13, 2012
  5. Jan 13, 2012 #4

    micromass

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    Yeah, it is connected, but not path connected. We were looking for something that wasn't path connected.
     
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