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Math Help - Q`s in topology need to be solve !

  1. #1
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    Post Q`s in topology need to be solve !

    Q1:-
    If ( X , T) & ( Y , J ) are both Hausdrff space`s , so is ( X x Y , P X x Y ) .

    remark:- P X x Y = the topology of the basis of ( X x Y )

    Q2:-
    A subspace of a Hausdrff space is also a Hausdrff space .


    Whith many many thnx
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  2. #2
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    If (x_1,y_1),(x_2,y_2)\in X\times Y pick open sets U_1,U_2\subset X and V_1,V_2\subset Y (all open in their respective topologies) such that x_i\in U_i and y_i\in V_i and U_1\cap U_2=\emptyset = V_1\cap V_2. What happens if (U_1\times V_1) \cap (U_2\times V_2)\neq \emptyset?

    Let x,y\in Y\subset X pick U,V \subset X such that x\in U, y\in V and U\cap V= \emptyset then what about U\cap Y and V\cap Y
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  3. #3
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    Quote Originally Posted by Jose27 View Post
    If (x_1,y_1),(x_2,y_2)\in X\times Y pick open sets U_1,U_2\subset X and V_1,V_2\subset Y (all open in their respective topologies) such that x_i\in U_i and y_i\in V_i and U_1\cap U_2=\emptyset = V_1\cap V_2. What happens if (U_1\times V_1) \cap (U_2\times V_2)\neq \emptyset?

    Let x,y\in Y\subset X pick U,V \subset X such that x\in U, y\in V and U\cap V= \emptyset then what about U\cap Y and V\cap Y
    first of all :- many many thnk`s for your help
    now
    the first section is 100% true

    but then ....???

    Let x,y\in Y\subset X pick U,V \subset X such that x\in U, y\in V and U\cap V= \emptyset then what about U\cap Y and V\cap Y[/QUOTE]

    how came Y be a subset of X ??!!!!!

    ___________________________________-

    now i think that , we begin with
    let (U_1\times V_1) \cap (U_2\times V_2)\ = h

    h \in  (U_1\times V_1) \   & h \in  (U_2\times V_2)\

    then same how we will get a contradiction ( how i don`t know ) ???
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  4. #4
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    ok i`ve got the prove
    and I`ll put it her soon
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