Talk:Quotient space

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Mathematics rating:	Start Class	Mid Priority	Field: Topology
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Maybe High-Priority. Geometry guy 22:36, 10 July 2007 (UTC)

can anyone tell me why its called "quotient"?

[edit] im f homeomorphic to X/~

I made a correction there--if f is not an open map, there is a continuous bijection from X/~ to im f, but it is not a homeomorphism.--Todd 15:07, 19 July 2006 (UTC)

You're right in thinking that a homeomorphism needs to be an open map. However, in the text the topology of Y is defined as the finest topology that makes f continuous: V is open in Y if and only if it's preimage under f is open in X. A topology on Y wasn't assumed; it was constructed. Originally, Y was only assumed to be a set. The construction not only makes f continuous, it also makes it open. I think the original text was right so unless I hear back from you soon, I'm going to change the article back. (Perhaps I'll try to clarify this point.) Lunch 18:38, 19 July 2006 (UTC)

Quotient maps aren't always open maps. However, the natural quotient map taking a space to an orbit space is an open map. 76.21.73.242 (talk) 22:32, 5 April 2008 (UTC)

[edit] Characterization of quotient maps

I made a correction in the following statement (change from "characterized by" to "characterized among surjective maps by"):

Quotient maps q : X → Y are characterized by the following property: if Z is any topological space and f : Y → Z is any function, then f is continuous if and only if fq is continuous.

Suppose the property holds for a map Failed to parse (Cannot write to or create math output directory): q:X\to Y . While q being continuous and Failed to parse (Cannot write to or create math output directory): V\subseteq Y

being open iff Failed to parse (Cannot write to or create math output directory): q^{-1}(V)
is open are quite easy to prove, I believe we cannot show q is onto. For suppose it isn't and on Y we have the strongest topology making q continuous (so on Y\q(X) the topology is discrete). Then the property holds (Failed to parse (Cannot write to or create math output directory): f^{-1}(U)
is open iff Failed to parse (Cannot write to or create math output directory): f^{-1}(U)\cap q(X)
is open iff Failed to parse (Cannot write to or create math output directory): q^{-1}(f^{-1}(U))=(fq)^{-1}(U)
is open, but Failed to parse (Cannot write to or create math output directory): q
is not a quotient map (since it is not surjective). --87.205.250.125 (talk) 21:56, 8 August 2008 (UTC)