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Another topology on the naturals

  1. Aug 27, 2010 #1

    radou

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    1. The problem statement, all variables and given/known data

    Let X = {1, 2, 3, 4, 5} and V = {{2, 3, 4}, {1, 4, 5}, {2, 4, 5}, {1, 3}} be a subbasis of a topology U on X.

    a) find all dense subsets of the topological space (X, U)
    b) let f : (X, U) --> (X, P(X)) be a mapping defined with f(x) = x (P(X)) is the partitive set of X). Is f continuous?

    3. The attempt at a solution

    a) Since V is a subbasis, then the family of all finite intersections of the elements of V form a basis B. So, B = {{2, 3, 4}, {1, 4, 5}, {2, 4, 5}, {1, 3}, {1}, {3}, {4}, {2, 4}, {4, 5}, {0}}. A subset D of X is dense in X iff every open set in X intersects D. So, D = {1, 3, 4}. To make sure this is right, we can see if Cl(D) = X. The closed sets in X are {{1, 2, 3, 5}, {1, 3, 5}, {1, 2, 3}, {2, 3, 4, 5}, {1, 5}, {2, 3}, {1, 3}, {2, 4, 5}, {1, 2, 4, 5}, {1, 2, 3, 4, 5}}. The intersection of all closed sets containing D is {1, 2, 3, 4, 5} (it's the only set containing D, btw), so Cl(D) = X.

    b) A mapping f between topological spaces is continuous iff the preimage of every open set in the codomain is open in the domain. For all sets in P(X) which are contained in U, this is obvious. If we take a set in P(X) (for example {2, 3}) which is not in U, then its preimage is the empty set, and hence is open in U. So, f is continuous.

    Thanks in advance.
     
  2. jcsd
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