Quotient Topology Explained: A Primer for Beginners

[shankarvn]

Can someone explain the concept of Quotient topology. I tried to read it from a book on topology by author "James Munkres" . It was okay but I did not get a feel of what he was trying to do.. He talks about cutting and pasting elements. I kind of got lost in that.. If someone could give me a physical picture, it would be great.. I believe this is an important concept. Looking forward to your replies.
Shankar

 

[mathwonk]

I have not used this concept too much, but I recall a little from reading kelley, general topology.

take any topological space X and any surjective map X-->Y to a set Y. then this map will be continuous with certain topologies on Y and not continuous with others.

i.e. the map is continuous iff every open set in Y ahs an open inverse image. so the more open sets Y has, the harder it is for the map to be continuous. E.g. if Y has the smallest possible topology, with only the empty set and Y open, the map is always continuous.

suppose we give Y the largest number of open sets, such that the map is still continuous. this i believe is called the quotient topology on Y.

i.e. given any surjective map X-->Y, we can view Y as made by taking a "quotient" of X by an equivalence relation.


just consider two points of X equivalent if they have the same image under the map.

Then given such a quotient space Y of X, the points of Y are actually subsets of X. Then the quotient topology on Y is expressed as follows: a set in Y is open iff the union in X of the subsets it consists of, is open in X.


This is a basic but simple notion. important, but nothing deep here except the idea of continuity, and the general idea of enhancing the structure of a set of equivalence classes.

the same thing goes on everywhere. given a group and a normal subgroup, one gets an equivalence relation on the elements of the group and makes a group out of these classes.

given a ring and an ideal one makes a ring out of the quotient object.

in general quotient constructions are techniques for constructing "maximal" targets for surjective maps subject to a certain condition.


I'm sorry. you asked for a physical picture. imagine the real line mapping to the circle by the map taking t to (cos(t),sin(t)). the "wrapping map" or polar coordinates map. then try to convince yourself that the quotient topology is exactly the usual topology on the circle.

 

[shankarvn]

Thank you sir. That helps. Can you throw some more light on the wrapping example that you gave in the last line. I believe everything in mathamatics tries to capture some physically intuitive idea. So I guess there should be more to this idea. I guess most geometries are obtained by cutting, wrapping and pasting exisitng geometries. Can you tell me more about this.(since it looks important to me)?Just an example will do. Thanks again.

 

[mathwonk]

have you seen polar coordinates? one measures angles on a circle by radians. i.e. t radians corresponds to a certain point on the circle. so if we have a straight t - axis, then sending 0 to the point (1,0) on the circle, and sending the point on the t axis, at distance t from the origin, to the point on the unit circle whose arclength from (1,0) has length equal to t, is called the polar coordinate map, or wrapping map.

think of wrapping a straight thread around a circular spool.

in coordinates it is given by cosine and sine. i.e. we map the t- line onto the unit circle by sending the point with coordinate t on the t - line, to the point with coordinates (cos(t), sin(t)) on the unit circle.

somehow i feel i am not understanding your question. usually in this situation someone else does, and steps in and helps out. maybe they will.

 

[shankarvn]

Thank you sir
Shankar

 

[mathwonk]

mr shankar you are a gentleman and a scholar.

 

[talolard]

Maybe a few years late to the party, but that's an excellent explanation. ThankYou.