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[SOLVED] Quotient Maps

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,104
The Astral plane
First a few definitions from my text in case the language is not quite universal.

1. Let X and Y be topological spaces; let $p: X \to Y$ be a surjective map. The map p is said to be a quotient map, provided a subset U of Y is open in Y if and only if $p^{-1}(U)$ is open in X.

2. If a set X is a (topological) space and if $p: X \to Y$ is a surjective map, then there exists exactly one topology T on Y relative to which p is a quotient map; it is called the quotient topology induced by p.

3. Let X be a (topological) space, and let Y be a partition of X into disjoint subsets whose union is X. Let $p: X \to Y$ be the surjective map that carries each point of X to the element of Y containing it. In the quotient topology induced by p the space Y is called a quotient space of X.

Now for a few questions. I am addressing the definitions "backward", which is always a bad idea.

Definition 3 is clearly referring to an equivalence relation ~ on X, which should indicate that Y is the set $\{y | y \sim x \} ~ \forall ~ x \in X$. Taking this as a given, definition 2 seems to be merely saying that there is more than one possibility of a quotient topology on Y since there is in general more than one equivalence relation from X to Y and hence more than one p which would induce different quotient spaces Y.

But given all of this definition 1 is kicking my butt. I note that neither definitions 1 and 2 require that p must be an equivalence relation (as definition 3 states) but I just can't get get my mind off the equivalence relation concept to generate the quotient space. Is it possible to have a quotient map p that generates a quotient topology on Y but that does not require p to be an equivalence relation (and hence Y is not a quotient space?)

Any comments?

Thanks,

-Dan
 

castor28

Well-known member
MHB Math Scholar
Oct 18, 2017
238
Brussels, Belgium
Hi,

Note first that case 3 is a particular case: $Y$ is a set of subsets of $X$. In cases 1 and 2, the spaces $X$ and $Y$ are unrelated.

In particular, case 2 does not require $p$ to arise from an equivalence relation.

However, there is really no difference: any surjective map $p:X\to Y$ between sets defines an equivalence relation $\approx$ on $X$ by $x_1\approx x_2$ if $p(x_1)=p(x_2)$. This allows you to define a quotient set of $X$ irrespective of any topological concept.
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,104
The Astral plane
Alright. I think I'm trying to think this one too hard. Thanks for the comment about a surjective functions and equivalence relations. I had never put the two together, though in retrospect it seems obvious.

For some reason I was requiring that the last two definitions were part of the definition of a quotient map. The other two definitions clearly are not referring to quotient maps but definitions about where we can take things when we do have a quotient map.

Thanks for the help!

-Dan