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Error in a book?

  1. Aug 10, 2010 #1

    I'm reading through a book and have come across something that seems like an obvious error to me. The books says

    Now it's been a long time since I've done any topology, but isn't this the box topology rather than the product topology? I just want to make sure I'm not going crazy.
  2. jcsd
  3. Aug 10, 2010 #2
    I think the product topology and box topology coincide for finite products. If you are looking at the space X x Y in particular, then the product topology is generated by the set

    [tex]\{ O_X \times Y :O_X \in T_X \} \cup \{X \times O_Y:O_Y \in T_Y \} [/tex]

    Finite intersections of these sets are open as well. In particular,

    [tex](O_X \times Y) \cap (X \times O_Y) = O_X \times O_Y[/tex]

    is open.
  4. Aug 10, 2010 #3

    I think you're right. Planetmath says that they coincide when the index set is finite.
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