Results 1 to 2 of 2

Thread: Product topology

  1. #1
    Newbie
    Joined
    Jan 2009
    Posts
    13

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    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 $\displaystyle (X,T_1)$ and $\displaystyle (Y, T_2)$ be topological spaces. A basis element of the product topology on $\displaystyle A \times B$ has the form $\displaystyle V_1 \times V_2$, where $\displaystyle V_1, V_2$ be open sets in the subspace topology on $\displaystyle (A,S_1)$, and $\displaystyle (B,S_2)$, respectively.

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

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

    The converse is similar to the above.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. R^w in the product topology?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 2nd 2011, 03:11 PM
  2. Product Topology
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Feb 10th 2010, 04:01 PM
  3. Product topology
    Posted in the Differential Geometry Forum
    Replies: 8
    Last Post: Mar 4th 2009, 11:35 AM
  4. discrete topology, product topology
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Dec 13th 2008, 02:19 PM
  5. discrete topology, product topology
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Dec 7th 2008, 01:01 PM

Search Tags


/mathhelpforum @mathhelpforum