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Union of sets | quick question

  1. Jun 3, 2010 #1
    1. Suppose open sets [tex]V_{\alpha}[/tex] where [tex] V_{\alpha} \subset Y \: \forall \alpha [/tex], is it true that the union of all the [tex]V_{\alpha}[/tex] will belong in Y? (i.e. [tex]\bigcup_{\alpha} V_{\alpha} \subset Y[/tex])

  2. jcsd
  3. Jun 3, 2010 #2


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    Of course it's true. If you aren't sure, I think you'd better try and prove it.
  4. Jun 4, 2010 #3


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    Let x be an element of that union. Then what must be true about x?
  5. Jun 4, 2010 #4
    Ok, if x is a member of [tex]\bigcup_{\alpha} V_{\alpha}[/tex] then x is a member of [tex]V_{\alpha}[/tex] for some [tex]\alpha[/tex]. But [tex] V_{\alpha} \subset Y \: \forall \alpha [/tex]. Then x is also an element of Y. Since this is true for every x in [tex]\bigcup_{\alpha} V_{\alpha}[/tex], then it must be the case that [tex]\bigcup_{\alpha} V_{\alpha} \subset Y \: \forall \alpha[/tex].

    Was that convincing?
  6. Jun 4, 2010 #5
  7. Jun 4, 2010 #6
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