(Throughout this post, I am talking about vector spaces.)
I had the pleasure of doing Abstract Algebra two semesters early, however, I feel like some general context was lost in the process. While I have a certain intuitive understanding of both quotient groups and cosets, I had never heard the terms "quotient space" or "equivalence classes". I am trying to create an intuitive notion of quotient spaces, based on my knowledge of quotient groups.
Consider the group $\mathbf{Z}\times\mathbf{Z}$, and its normal subgroup $\langle(1,0)\rangle$. We see that the quotient group $G/N$ is isomorphic to $\mathbf{Z}$. In other words, it is still a group. In particular, we can read this as $G \:\text{mod}\:N$, which gives an even more intuitive notion of a quotient groups. (And helps a lot when working with polynomial rings.)
Do quotient spaces and equivalence classes follow in an obvious way? Is there an example, of the same simplicity as the example provided above that can be used to explain the connection?
Best Answer
If $R$ is an equivalence relation on a set $X$, we naturally get the quotient set $X/R$ whose elements are the equivalence classes under $R$. Almost all constructions of "quotient spaces" (of vector spaces, abelian groups, whatever) are then as follows: define the structure on $X/R$ so as to make the canonical map $X\rightarrow X/R$ (which takes every element to its equivalence class) a morphism.
An example which you should have seen: if $G$ is a group and $N$ a normal subgroup, define an equivalence relation $R$ on $G$ by $xRy$ iff there is $n\in N$ with $y=nx$. The equivalence classes are exactly the cosets, and the quotient set is denoted $G/N$ and is the quotient group when its group structure is defined so as to make $G\rightarrow G/N$ a homomorphism.