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Topology Question

  1. Nov 14, 2007 #1
    1. The problem statement, all variables and given/known data
    I am asked to show that T=[particular point topology on R^2 ((0,0) being the particular point)] is equal to T'=[topology on R^2 from taking the product of R in the particular point topology (0 being particular point) with itself].



    3. The attempt at a solution

    I'm pretty sure the book is wrong and this problem can't be solved. Consider the set {(0,0),(1,1)}. This is open in T, but I cannot see how this is open in T' since for the point (1,1) we cannot construct an open set that contains (1,1) but is contained in {(0,0),(1,1)}. The closest we can get is {0,1}x{0,1}={(0,0),(1,0),(0,1),(1,1)}. Am I correct, or am I missing something?
     
  2. jcsd
  3. Nov 14, 2007 #2

    HallsofIvy

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    For my own benefit, a set is open in the "particular point topology" if and only if it is either empty of contains the "particular point". In particular, a set in R^2 with the particular point topology, (0,0) being the "particular point", T, if and only if it contains (0,0).
    A set is open in the "Cartesian product topology" if and only if it is the Cartesian product of two open sets. Here a set is open in T' if and only if it contains a pair (0,y) and the pair (x, 0).



    If I have understood this correctly, yes you are right!
     
  4. Nov 14, 2007 #3
    Not quite, the "Cartesian product topology" is generated from the set of Cartesian products of two open sets. If we took your definition for open sets in R^2, then it would be possible for the union of two open sets to not be open. See the difference?
     
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