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Thread: Quotient Maps

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    Forum Admin topsquark's Avatar
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    Quotient Maps

    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
    Last edited by topsquark; Feb 12th 2018 at 02:16 PM.
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    Re: Quotient Maps

    Quote Originally Posted by topsquark View Post
    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.
    Dan, I am a long way from any research in topology. When I was active it in Moore Spaces but once I did read on Quotient Maps. I found the book General Topology by Steven Willard helpful. Begin on p58 section 9 (I hate this text for its section numbering) . I remember very little. Nevertheless I gladly offer to help, even if it comes to naught.
    Thanks from topsquark
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    Forum Admin topsquark's Avatar
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    Re: Quotient Maps

    Quote Originally Posted by Plato View Post
    Dan, I am a long way from any research in topology. When I was active it in Moore Spaces but once I did read on Quotient Maps. I found the book General Topology by Steven Willard helpful. Begin on p58 section 9 (I hate this text for its section numbering) . I remember very little. Nevertheless I gladly offer to help, even if it comes to naught.
    Thanks, I appreciate that. I am in the market for a higher level (graduate level) text on Topology, but quotient maps aren't the only concept I'm having troubles with. A good undergrad text would not be a waste...Quite the contrary. Would you recommend "General Topology" to be in that category?

    Thanks!

    -Dan
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    Re: Quotient Maps

    Quote Originally Posted by topsquark View Post
    Thanks, I appreciate that. I am in the market for a higher level (graduate level) text on Topology, but quotient maps aren't the only concept I'm having troubles with. A good undergrad text would not be a waste...Quite the contrary. Would you recommend "General Topology" to be in that category?
    Dan, some how I did not see your reply until today.
    Introduction To General Topology by Helen Cullen is a bit dated but still the best text for beginning graduate level in my opinion.
    Her section on Quotient Spaces (p91) does match the definitions you posted. They may be equivalent, I have not tried to show it.
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