Example in topology: quotient maps and arcwise connected

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Discussion Overview

The discussion revolves around the properties of quotient maps in topology, specifically examining an example involving the topologist's sine curve. Participants explore whether this example satisfies certain conditions related to pathwise connectivity and overall connectivity.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant proposes an example of a quotient map p: X → Y, where X is the topologist's sine curve and Y is the interval [0,1], asserting that Y is pathwise connected and that the preimages under p are also pathwise connected.
  • Another participant agrees with the initial proposal, indicating that it seems acceptable.
  • Several participants challenge the claim about the topologist's sine curve, arguing that it is connected, as it is the closure of the continuous image of the semi-open interval (0,1].
  • One participant reiterates that while the topologist's sine curve is connected, it is not path connected, suggesting that the original example does not meet the criteria being sought.

Areas of Agreement / Disagreement

Participants express disagreement regarding the connectivity of the topologist's sine curve, with some asserting it is connected and others emphasizing its lack of path connectivity. The discussion remains unresolved regarding the initial example's validity as a quotient map under the specified conditions.

Contextual Notes

There are conflicting interpretations of the properties of the topologist's sine curve, particularly concerning its connectivity and path connectivity. The discussion highlights the need for clarity on definitions and the implications of these properties in the context of quotient maps.

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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.
 
Last edited:
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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