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walters

  • 2 years ago

Show that {x} are open sets in X for all points x∈X, then all subsets of X are also open in X.

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  1. experimentX
    • 2 years ago
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    {x} collection of open sets ... open covering of X?

  2. walters
    • 2 years ago
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    yes

  3. experimentX
    • 2 years ago
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    looks like i am not understanding the Q ... you sure the Q is right?

  4. walters
    • 2 years ago
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    yes it is like that

  5. experimentX
    • 2 years ago
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    any background on the Q? for some reason I find point set topology very hard.

  6. walters
    • 2 years ago
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    what i know that the set is open when all its point are interior

  7. experimentX
    • 2 years ago
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    But discrete points are itself closed set.

  8. experimentX
    • 2 years ago
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    where does this problem come from (book) ??

  9. walters
    • 2 years ago
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    assignment

  10. experimentX
    • 2 years ago
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    which book are you using as Text Book ??

  11. walters
    • 2 years ago
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    RG Bartle and DR Sherbert ,introduction to real analysis john wiley &sons

  12. experimentX
    • 2 years ago
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    @walters have you managed to go all though chapters to reach Point Set Topology at the end?

  13. walters
    • 2 years ago
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    not really but i think his question is open and closed sets

  14. walters
    • 2 years ago
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    i understand it this way but i can't show it |dw:1361566625628:dw|

  15. walters
    • 2 years ago
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    |dw:1361567087644:dw|

  16. experimentX
    • 2 years ago
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    Let F be collection of A's ... and \( X = \cup_{A \in F} A\) and \( B(x;r) \subset A_i \forall x \in A, \implies B(x;r) \subset X\) shows that X is open set.

  17. experimentX
    • 2 years ago
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    is X compact??

  18. walters
    • 2 years ago
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    "compact "what do u mean

  19. experimentX
    • 2 years ago
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    I think what you are doing is Let X be a closed set, and \( x \in X \) be a point in X. and \( X - {x}\) is a subset of X which is open in X. let me ask others.

  20. experimentX
    • 2 years ago
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    X-{x}

  21. walters
    • 2 years ago
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    is what i am thinking "but not quit sure"

  22. experimentX
    • 2 years ago
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    let me ask experts ... I'll reply you in sometime.

  23. walters
    • 2 years ago
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    i think since the subset of X are open this also implies that {x}and its complements are also open

  24. experimentX
    • 2 years ago
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    i got the hint: in the discret topology all points are open

  25. walters
    • 2 years ago
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    so does this mean that there is no relationship between {x} and its complement

  26. experimentX
    • 2 years ago
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    no ... use the fact that union of infinite number of open sets is and open set.

  27. experimentX
    • 2 years ago
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    or simply just union of open sets.

  28. walters
    • 2 years ago
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    i mean like "ie if {x} is open it does not imply that its complement is also closed "

  29. walters
    • 2 years ago
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    is it possible ?

  30. experimentX
    • 2 years ago
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    I guess not in discrete topology

  31. experimentX
    • 2 years ago
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    |dw:1361569963336:dw| usually we have neither closed nor open set.

  32. experimentX
    • 2 years ago
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    but in discrete topology we seem to have http://mathworld.wolfram.com/DiscreteSet.html http://mathworld.wolfram.com/DiscreteTopology.html

  33. experimentX
    • 2 years ago
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    the neighborhood of the boundary will not contain any point of that inner set, which makes the compliment open.

  34. walters
    • 2 years ago
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    ok i get ur statement

  35. experimentX
    • 2 years ago
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    as to your question, the subsets of X will be power sets of its elements which are open. And the union of open set is open. Hence all the subsets will be open.

  36. experimentX
    • 2 years ago
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    P.S. I am not sure, I don't have experience with Topology.

  37. walters
    • 2 years ago
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    @phi

  38. phi
    • 2 years ago
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    I am an engineer, and have not studied analysis.

  39. walters
    • 2 years ago
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    @JamesJ and @KingGeorge

  40. walters
    • 2 years ago
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    |dw:1361653147330:dw| i think they mean it this way because of the question

  41. walters
    • 2 years ago
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    @jacobian

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