Example in topology: quotient maps and arcwise connected

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The discussion centers on the properties of quotient maps in topology, specifically using the topologist's sine curve as an example. The proposed map p: X → Y projects the sine curve onto the x-axis, resulting in Y being pathwise connected. However, it is clarified that while the topologist's sine curve is connected, it is not path connected, which contradicts the initial requirement. The participants confirm that the sine curve serves as a standard example of a space that is connected but lacks path connectivity. This highlights the distinction between different types of connectivity in topological spaces.
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Just to make sure that I'm not overlooking anything, is the following an example of a quotient map p: X \to Y with the properties that Y is pathwise connected (i.e. connected by a continuous function from the unit interval), \forall y \in Y: p^{-1}(\{ y \}) \subset X also pathwise connected and such that X is not connected?

As X 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 Y = [0,1] 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 \forall y \in [0,1]: p^{-1}(\{ y \}) is also pathwise connected.
 
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Seems ok! :smile:
 
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.
 
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Bacle2 said:
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.

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

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