1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Show If a point A is in the interior, then it has a neighborhood contained in A.

  1. Sep 2, 2012 #1
    1. The problem statement, all variables and given/known data
    Let A be a topological space and let A[itex]\subseteq[/itex]X be any subset.
    Show: If a point A is in the interior, then it has a neighborhood contained in A.


    2. Relevant equations
    Neighborhoods are defined to be open in my book.
    Int(A) = [itex]\bigcup[/itex]{C[itex]\subseteq[/itex]A and C is open in X}


    3. The attempt at a solution
    Let p[itex]\in[/itex]Int(A).
    Then p[itex]\in[/itex][itex]\bigcup[/itex]{C[itex]\subseteq[/itex]X:C[itex]\subseteq[/itex]A and C is open in X}
    So, [itex]\exists[/itex] an open set C' s.t. p[itex]\in[/itex]C' and C[itex]\subseteq[/itex]A
    Q.E.D.
     
  2. jcsd
  3. Sep 2, 2012 #2
    Sounds good!
     
  4. Sep 2, 2012 #3
    Thank you! Could you check my work backwards now?
    I can say:

    Let C'[itex]\subseteq[/itex]A be open and non-empty
    Let p[itex]\in[/itex]C
    Thus p[itex]\in[/itex][itex]\bigcup[/itex]{C[itex]\subseteq[/itex]X:X[itex]\subseteq[/itex]A and C is open in X}
    Thus p[itex]\in[/itex]Int A
     
  5. Sep 2, 2012 #4
    That's correct. But you got to clean up your presentation. What's C' for example??

     
  6. Sep 2, 2012 #5
    C' was meant to be one set that has that element and
    I meant to type let p[itex]\in[/itex]C'
    and I meant to write
    p∈⋃{C⊆X:C⊆A and C is open in X}

    Do I still need to clean up?
     
  7. Sep 2, 2012 #6
    Ah ok. I guess it's ok now!
     
  8. Sep 2, 2012 #7
    Thank you very much! If you have time, would you mind helping me with my other question? It's about the equivalence of a bounded set with a closed ball. I just want to check my proof one direction. Thanks again!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...