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anonymous

  • one year ago

help needed here please

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  1. anonymous
    • one year ago
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    Let X be a complete metric space and {On} is countable collection of dense open subset of X. Show that On is not empty.

  2. anonymous
    • one year ago
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    @Michele_Laino

  3. Michele_Laino
    • one year ago
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    I can solve that question using the Theorem of Category of Baire

  4. anonymous
    • one year ago
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    yes please use that theorem

  5. Michele_Laino
    • one year ago
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    ok!

  6. Michele_Laino
    • one year ago
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    Using that theorem, we can state this: since X is a complete metric space, then any countable collection of dense open subsets has dense intersection

  7. anonymous
    • one year ago
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    ok

  8. Michele_Laino
    • one year ago
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    now, let: \[\Large {O_n} = {\left\{ {{V_n}} \right\}_{n \in \mathbb{N}}}\] be our countable collection

  9. anonymous
    • one year ago
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    ok please continue , i am here

  10. Michele_Laino
    • one year ago
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    the subsequent set: \[\Large \overline {{ \cap _n}{V_n}} = X\]

  11. Michele_Laino
    • one year ago
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    since by the Baire's theorem the set: \[\Large {{ \cap _n}{V_n}}\] is dense in X

  12. anonymous
    • one year ago
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    is that all?

  13. Michele_Laino
    • one year ago
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    no, please let's suppose, by absurdum, that On is empty

  14. Michele_Laino
    • one year ago
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    that means each Vn is empty: \[\Large {O_n} = \emptyset \Rightarrow {V_n} = \emptyset ,\quad \forall n\]

  15. anonymous
    • one year ago
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    ok

  16. Michele_Laino
    • one year ago
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    so, also the intersection is empty: \[\Large { \cap _n}{V_n} = \emptyset \]

  17. Michele_Laino
    • one year ago
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    because the intersection of empty sets is also empty

  18. anonymous
    • one year ago
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    yap

  19. Michele_Laino
    • one year ago
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    now, the empty set is both a closed and open set, and we can write this: \[\Large \emptyset = \overline \emptyset = \overline {{ \cap _n}{V_n}} = X\] which is a contradiction, since by hypothesis, X is not empty

  20. Michele_Laino
    • one year ago
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    that's all!

  21. anonymous
    • one year ago
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    thanks . want to copy it and study but can i ask ine last question for now?

  22. Michele_Laino
    • one year ago
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    ok!

  23. anonymous
    • one year ago
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    Let (X, d) and (Y, d) be metric spaces and f a mapping of X into Y. Let τ1 and τ2 be the topologies determined by d and d1 respectively. Then f(X, τ) (y, τ) is continuous if and only if ; that is if x1, x2, . . . , xn, . . . , is a sequence of points in (X, d) converging to x, show that the sequence of points f(x1), f(x2), . . . , f(xn), . . . in (Y, d) converges to x.

  24. Michele_Laino
    • one year ago
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    please I have to go out now, I will come back between 2 hours

  25. anonymous
    • one year ago
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    @Michele_Laino

  26. Michele_Laino
    • one year ago
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    a proof can be this: let's suppose that f is contnuous, then the sequence x1, x2,...xn can be viewed as a continuous function from the set of natural number to the metric space (X,d), therefore, using the theorem of composition of continuous function, we can say that the function: \[\Large \mathbb{N} \to Y:n \to f\left( {{x_n}} \right)\] is also continuous and we have: \[\Large \mathop {\lim }\limits_{n \to \infty } f\left( {{x_n}} \right) = f\left( {{x_0}} \right)\]

  27. Michele_Laino
    • one year ago
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    vice versa, let's suppose that f is not a continuous function at x_0. Then we can find a neighborhood V around f(x_0) of Y, such that for each neighborhood U around x_0 of X we have: \[\Large f\left( U \right) \not\subset V\] We can choose each of Us like a ball of radius r= 1/n, n=1,2,3,.... Inside each of such balls U, we can find an element x_n such that: \[\Large f\left( {{x_n}} \right) \notin V\] in so doing we got a subsequence x1, x2,...,xn which converges to x_0, nevertheless f(x_n) doesn't converge to f(x_0), namely a contradiction

  28. anonymous
    • one year ago
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    ok

  29. anonymous
    • one year ago
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    is that all?

  30. Michele_Laino
    • one year ago
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    yes! we are done

  31. anonymous
    • one year ago
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    waw, thanks .

  32. Michele_Laino
    • one year ago
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    :)

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