Re: Example of a quotient map that is neither closed nor open

On Fri, 21 Dec 2007, hartkp@xxxxxxxxx wrote:
On Dec 21, 3:43 pm, TCL <tl...@xxxxxxx> wrote:
On Thu, 20 Dec 2007, TCL wrote:

Let X, Y be topological spaces. A continuous
surjective map f from X onto Y is called a quotient
map if E is open whenever f^{-1}(E) is open.

What is an example of a quotient map that is
neither open nor closed?

An example was given in Kelley's General Topology,
but that does not
make sense to me.

f:[0,2) -> S^2, t -> (sin 2pi.t, cos 2pi.t)

[0,1/2) open subset [0,2), f([0,1/2)) not open.
[3/2,2) closed subset [0,2), f([3/2,2)) not closed.

On what page is Kelley's example?

That is a good example. Kelley's example on page 95 is a mistake, I believe.

No it's not: Kelley mentions it as a map that is neither open nor
closed; there is no claim that the map is quotient.

Hm. You're right. p^-1(0) = { (x,y) | x = 0, y /= 0 }
is open set within x-axis \/ y-axis - origin.
.

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