What constitutes a quotient map?

[jhicks]

This is not directly a homework problem, so I opted not to place this question there. From what I have read/gathered from the internet/my textbook, a quotient mapping is any surjective, continuous mapping from a space X to a space comprised of the equivalence classes of all x in X from a relation ~. Is there anything more to it? I could find no one source that spelled out the definition for me, complete with all the conditions that must be satisfied to constitute one.

The question that prompted this was something to the effect of proving a particular mapping from an arbitrary space to a subset of that space was a quotient map, and I was given the information indirectly that it was an onto mapping that was continuous. I basically made up an equivalence relation that satisfied the definition of the particular mapping, and that seemed like enough.

 

[HallsofIvy]

This is not directly a homework problem, so I opted not to place this question there. From what I have read/gathered from the internet/my textbook, a quotient mapping is any surjective, continuous mapping from a space X to a space comprised of the equivalence classes of all x in X from a relation ~. Is there anything more to it? I could find no one source that spelled out the definition for me, complete with all the conditions that must be satisfied to constitute one.

The question that prompted this was something to the effect of proving a particular mapping from an arbitrary space to a subset of that space was a quotient map, and I was given the information indirectly that it was an onto mapping that was continuous. I basically made up an equivalence relation that satisfied the definition of the particular mapping, and that seemed like enough.

The most general definition of quotient map is that it maps a member of a set to the equivalence set it belongs to for some equivalence relation. The most general definition of "quotient map" doesn't require things like continuous because that is not defined for general sets. If you were able to find an equivalence relation so that the given mapping works with that equivalence relation, then, yes, it is a quotient map.

 

[ZioX]

Hmm, that definition is odd as it doesn't talk about what individual elements map to. Much too general for it's actual application.

But for the definition where it maps elements in a set to its equivalence class: the surjective requirement is redundant as equivalence relations partition up the sets. Continuity implies you have some kind of topology which is not necessarily needed -- this may depend on context.

If you're unfamiliar with equivalence relations or quotient maps you can try constructing concrete examples w/ very little work. Take some modulo congruency equivalence relation on the integers and consider the map that takes any element into its equivalence class.

 

[jhicks]

But for the definition where it maps elements in a set to its equivalence class: the surjective requirement is redundant as equivalence relations partition up the sets. Continuity implies you have some kind of topology which is not necessarily needed -- this may depend on context.

My definition was a little redundant - I realize that now - but only because I was trying to find the answer to the question I posed in the second paragraph and tried to considered what actually made a quotient map. Thank you for the clarifications though, Ziox and HallsofIvy

 

[fopc]

This is not directly a homework problem, so I opted not to place this question there. From what I have read/gathered from the internet/my textbook, a quotient mapping is any surjective, continuous mapping from a space X to a space comprised of the equivalence classes of all x in X from a relation ~. Is there anything more to it? I could find no one source that spelled out the definition for me, complete with all the conditions that must be satisfied to constitute one.

The question that prompted this was something to the effect of proving a particular mapping from an arbitrary space to a subset of that space was a quotient map, and I was given the information indirectly that it was an onto mapping that was continuous. I basically made up an equivalence relation that satisfied the definition of the particular mapping, and that seemed like enough.

Here's an example of how a "quotient" map might be constructed.

Let f:A->B (f an arbitrary map, A and B arbitrary sets). The map f induces a natural partition on its domain, A.
The equivalence relation associated with this partition is defined by, a~b iff f(a)=f(b).

Take ~ as your equivalence relation on A.

Then the map g:A->A/~ defined by the rule g(a)=[a]~ is sometimes referred to as the canonical map from A to the quotient set A/~.
It could also be called the quotient map wrt. ~. However, see link below for a different naming convention.

For the additonal properties you cited (e.g., continuity), see the link for details.

http://en.wikipedia.org/wiki/Quotient_map

 

[math8]

I think a quotient map is an onto map p:X-->Y (where X and Y are topological spaces) such that
U is open/closed in Y iff (p)-1(U) is open/closed in X.

 

[HallsofIvy]

I think a quotient map is an onto map p:X-->Y (where X and Y are topological spaces) such that
U is open/closed in Y iff (p)-1(U) is open/closed in X.

Did you not read the other responses? That is correct IF you are talking about quotient maps on topological spaces. However, the original question did not restrict to topology. Given a normal subgroup, H, of a group, G, we can define the "quotient" group G/H and there is no topology there.
Say that two elements, x1 and x2, of X are 'equivalent' if and only if p(x1)= p(x2) any your quotient space is homeomorphic to the topology space induced on the equivalence classes in X.