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Homework Help: Topology Proof

  1. Apr 21, 2010 #1
    1. The problem statement, all variables and given/known data


    show that T:=(A subset X |A = 0 or X\A is finite) is a topology on X,
    2. Relevant equations

    We need to show 3 conditions.

    1: X,0 are in T
    2: The union of infinite open set are in T
    3: The finite intersections of open sets are open.



    3. The attempt at a solution

    We see that [itex]A \subset X [/itex] is open in (T1 space) asX\A is finite

    To show condition 1
    if A = 0 the empty set it is in T
    and A\X = X than it is in T.


    To show 2

    let [itex]A \subset X [/itex] open in T1 as X\A is finite

    Let [itex]\alpha \in I [/itex] be an indexing set, [itex] A_\alpha \in T [/itex] so that [itex] A \subset X [/itex] be open as X\A is finite.

    Than the [itex]\cup_{\alpha \in I} [/itex] X\[itex]A_\alpha = \cap _{\alpha \in I}[/itex] (X\[itex]A_\alpha[/itex])

    Either each of the sets ( X\[itex]A_\alpha[/itex]) = X , in which case the intersection is all of X, or at least one of them is finite , in which case the intersection is a subset of a finite set and hence finite.

    To show 3

    Let [itex]A_1,A_2,A_3...A_n \subset X [/itex]be open as X\A is finite or all of X.

    To show that [itex] \cap A_{n} \in T [/itex]we must show that [itex]\cap[/itex] X\[itex]A_n[/itex] is either finite or all of X.

    But [itex]\cap X[/itex]\[itex]A_{n} = \cup X[/itex]\[itex]A_{n}[/itex].

    Either this set is a union of finite sets and hence finite, or for some X\[itex]A_{i} i \in I = X [/itex]and the union is all of X.


    Thus (A,T) is a topological space.
     
  2. jcsd
  3. Apr 21, 2010 #2
    You have the right idea for all of the conditions. Applying the set theory properties was the key.

    Have you heard of the countable complement topology?
     
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