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Math Help - Product topology

  1. #1
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    Product topology

    Let X and Y be the topological spaces, ans assume that A⊂X and B⊂Y. Then the topology on AxB as a subspace of the product XxY is the same as the product topology on AxB, where A has the subspace topology inherited from X and B has the subspace topology inherited from Y.
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  2. #2
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    Quote Originally Posted by horowitz View Post
    Let X and Y be the topological spaces, ans assume that A⊂X and B⊂Y. Then the topology on AxB as a subspace of the product XxY is the same as the product topology on AxB, where A has the subspace topology inherited from X and B has the subspace topology inherited from Y.
    Let (X,T_1) and (Y, T_2) be topological spaces. A basis element of the product topology on A \times B has the form V_1 \times V_2, where V_1, V_2 be open sets in the subspace topology on (A,S_1), and (B,S_2), respectively.

    Since S_1=\{A \cap U \text{ } | \text{ }U \in T_1\} and S_2=\{B \cap V \text{ } | \text{ } V \in T_2\}, we have V_1 = A \cap U_1 and V_2 = B \cap U_2, where U_1, U_2 be open sets in (X, T_1) and (Y, T_2).

    Since V_1 \times V_2 = ((A \cap U_1) \times (B \cap U_2)), we have a basis element of a topology on A \times B as a subspace of the product X \times Y, which is ((A \cap U_1) \times (B \cap U_2))=((A \times B) \cap (U_1 \times U_2)), where U_1, U_2 be open sets in (X, T_1) and (Y, T_2).

    The converse is similar to the above.
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