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Description of a quotient map

  1. Apr 12, 2014 #1
    I am reading munkres topolgy and I am struggling with understanding the following sentence:

    "We say that a subset C of X is saturated (with respect to the surjective map p:X→Y) if C contains every set p-1({y}) that it intersects"

    if you have the second edition its in chapter 2 section 22 (page 137)

    It's not that I have questions on it I just cant seem to make heads or tails of that sentence.

    any help would be appreciated
     
    Last edited: Apr 12, 2014
  2. jcsd
  3. Apr 12, 2014 #2
    All it means is that if C intersect p-1({y}) is nonempty, then C actually contains all of p-1({y}). So if C is saturated and p-1({y}) has say, two elements, it is not possible that only one of those elements is in C.

    This is equivalent to "C is a saturated subset of X if C is the preimage of some subset of Y".
     
  4. Apr 13, 2014 #3
    thanks

    do we make any demands on the subset of Y, need it be open, closed... or will any subset do?
     
  5. Apr 13, 2014 #4
    Any subset will do.
     
  6. Apr 13, 2014 #5

    micromass

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    Another equivalent form is that ##C## is saturated if and only if ##C=p^{-1}(p(C))##. So the exact form of the subset of ##Y## is ##p(C)##.

    That said, I prefer dealing with quotient mappings in terms of equivalence relations. Thus a quotient map ##p:X\rightarrow Y## induces a equivalence relation on ##X## given by ##x\sim x^\prime## if and only iff ##p(x) = p(x^\prime)##. The set of all equivalence classes can then be identified wuth ##Y##.

    In that form, we can give new equivalent forms of saturated sets. One such form is to say that a set ##C## is saturated iff it is the union of equivalence classes. Another form is to say that ##C## is saturated if for any ##x\in C## and ##y\in C## such that ##x\sim y## holds that ##y\in C##.

    All of these forms are easily seen to be equivalent, but sometimes one equivalent form might give more insight than another one.
     
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