Example in topology: quotient maps and arcwise connected

[nonequilibrium]

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.

 

[micromass]

Seems ok!

 

[Bacle2]

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.

 

[micromass]

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.

 

[blue_raver22]

Yes, your example is a valid one. It satisfies all the properties mentioned and serves as a good illustration of quotient maps and arcwise connectedness in topology.

The topologist's sine curve is a classic example often used in topology to demonstrate certain concepts. It is a subset of the plane that is not connected, but its closure is. This makes it a good candidate for your example as X.

The projection map p is a standard example of a quotient map. It maps the topologist's sine curve onto the interval [0,1] with the quotient topology. This topology is indeed the same as the usual topology on [0,1].

Furthermore, as you pointed out, Y is pathwise connected and p^{-1}(\{y\}) is also pathwise connected for all y \in Y. This can be easily verified by considering the continuous function f:[0,1] \to Y defined by f(t) = t, which connects any two points in Y.

Overall, your example serves as a good illustration of how quotient maps preserve certain topological properties, such as arcwise connectedness, and how they can be used to construct new spaces with interesting properties.