Question about some terminology

navigator

Why the mathematician call identification topology "quotient topology"?from my viewpoint,identification topology bear an analogy with difference(minus),not quotient(divide)...BTW,Could anybody tell me the reason why the mathematicians who coined the term "ring","ideal","soul" called them like that,is there any anecdote about this terminology...tell me sth about them...Thx...

navigator

Does anyone know about it?

HallsofIvy

I'm afraid I don't see what your problem is with "quotient". It is, of course, in analogy to "quotient groups". If the groups happened to be written in "additive notation", i.e. a+ b rather than ab, identity= 0 rather than 1, the quotient group might correspond to a "subtrahend group" but that it just cosmetic.

As for the other terms, yo might find this websit interesting:
http://members.aol.com/jeff570/mathword.html

navigator

Thanks for your reply.Well,what I doubt will be stated in the following.From the viewpoint of intuition,identification is a gluing operation.As an example,we could get cylinder from rectangle via identification IxI/~(where (0,y)~(1,y)),and we may see that the operation is like to subtract the edge ((0,y) or (1,y)) from the rectangle,so identification should have an analogy with subtraction，rather than division.and when we call it quotient topology,what is the relation between quotient topology and product topology?inversion or other?...