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How to show induced topological space

  1. I am beginning to read about the topology,
    I met a problem puzzled me for a while.

    If [tex]Y[/tex] is a topological space, and [tex]X\subset Y[/tex], we can make the set [tex]X[/tex] to be a topological space by defining the open set for it as [tex]U\cap X[/tex], where [tex]U[/tex] is an open set of [tex]Y[/tex].

    I would like to show that this indeed defines a topological space. But I failed to prove that there is the open set [tex]X[/tex] among those open sets defined above, i.e. [tex]U\cap X[/tex]. Anybody helps me?

    Otherwise, we can easily see that
    [tex] (U_1\cap X) \cap (U_2 \cap X) = (U_1\cap U_2)\cap X [/tex]
    and
    [tex] (U_1\cap X) \cup (U_2\cap X) = (U_1\cup U_2) \cap X [/tex]
    and
    [tex] \phi = \phi \cap X [/tex]
    And I lack the final piece that the [tex]X[/tex] is contained in the collection of open sets of [tex]X[/tex] defined above.

    Thanks!
     
  2. jcsd
  3. George Jones

    George Jones 6,388
    Staff Emeritus
    Science Advisor
    Gold Member

    Let U be Y.
     
  4. aha! How stupid I was!
    Just let [tex] U[/tex] be [tex] Y [/tex].
    Thank you a lot. :shy:
     
  5. micromass

    micromass 18,742
    Staff Emeritus
    Science Advisor
    Education Advisor

    Also, you might want to use [ itex] instead of [ tex ] if you don't want to start a new line every time...
     
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