Proof: Topology of subsets on a Cartesian product

  1. The problem statement, all variables and given/known data

    Let Tx and Ty be topologies on X and Y, respectively. Is T = { A × B : A[itex]\in[/itex]Tx, B[itex]\in[/itex]Ty } a topology on X × Y?

    The attempt at a solution

    I know that in order to prove T is a topology on X × Y I need to prove:
    i. (∅, ∅)[itex]\in[/itex]T and (X × Y)[itex]\in[/itex]T
    ii. T is closed under finite intersections
    iii. T is closed under arbitrary unions

    In order to prove (i) I would have to prove that ∅[itex]\in[/itex]A and ∅[itex]\in[/itex]B. I think this is true because the empty set is in all sets.
    I'm not sure how to approach proving that X[itex]\in[/itex]A as even though A[itex]\in[/itex]Tx, this implies that A[itex]\in[/itex]X or A is X. I'm not sure how continue from here. Same with Y[itex]\in[/itex]B.

    For ii. I think that since Tx and Ty are topologies themselves, they are closed under finite intersections, and since A[itex]\in[/itex]Tx and B[itex]\in[/itex]Ty then A and B are also closed under finite intersections, thus T is closed under finite intersections. I have to go more into detail with this but I just want to make sure if this is the right idea.

    I think iii. could also be proved with a similar argument to the one used to prove ii.
     
  2. jcsd
  3. For (i), your topology T is the set of all open sets A x B such that A is an element of T_x, and B is an element of T_y. X is an element of T_x, as it is required to be one by the same rules we a re trying to prove, as well as the empty set, and vice versa for Y being an element on T_y. Use this fact to show that X x Y is in your topology T.
     
  4. Fredrik

    Fredrik 10,545
    Staff Emeritus
    Science Advisor
    Gold Member

    This one should say ##\emptyset\in T## and ##X\times Y\in T##.

    It's not. It's a subset of all sets, but most sets don't have it as a member.

    It doesn't make sense to try to prove that X is a member of A when you haven't specified what A is.

    ##A\in T_x## doesn't imply what you say it implies. It just means that A is an open subset of X.

    Note that the definition of T says that T consists of of all cartesian products of two open subsets of X and Y, such that the first set is a subset of X and the second a subset of Y. To check if ##\emptyset\in T##, you should ask yourself if ∅ can be expressed as a cartesian product at all. X×Y is obviously a cartesian product, so to prove that X×Y is in T, you only have to prove...what?

    I don't see what A and B being closed under finite intersections have to do with anything. You need to start with a statement like "Let n be an arbitrary positive integer, and let ##E_1,\dots,E_n## be arbitrary members of T". Then you prove that ##\bigcap_{k=1}^n E_k\in T##.
     
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