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Is the product of two open sets open?

  1. Sep 25, 2008 #1
    Say we have open sets from two topological spaces. A is an open subset of T1 and B is an open subset of T2. So for these two open sets, A, B. Is A X B open itself? I see this is the case in R x R, where R is the real number line. I am wanting to say that this is just true in general... If so, anyone know a good proof? Or can anyway tell me how to show this?
     
  2. jcsd
  3. Sep 25, 2008 #2
    The answer depends on what topology you give onto the set [tex]T_1\times T_2[/tex].

    For example, if you let [tex]\mathbb{R}[/tex] have the standard topology, but give [tex]\mathbb{R}\times\mathbb{R}[/tex] a trivial topology [tex]\{\emptyset, \mathbb{R}\times\mathbb{R}\}[/tex], then [tex]]0,1[\times ]0,1[[/tex] is not open.

    A standard choice for the topology of [tex]T_1\times T_2[/tex] is so called product topology. Here's the Wikipedia page of it: http://en.wikipedia.org/wiki/Product_topology
    In product topology, [tex]A\times B[/tex] is open in [tex]T_1\times T_2[/tex] always when [tex]A[/tex] is open in [tex]T_1[/tex] and [tex]B[/tex] is open in [tex]T_2[/tex]. This is almost by definition of the product topology. However, the precise definition is slightly complicated, so there is something left to be proved.
     
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