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Thread: Closure Problem

  1. #1
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    Closure Problem

    Hello everyone.

    I am trying to prove that if $\displaystyle A \subseteq X$ is dense then for any open $\displaystyle T$ we have $\displaystyle T \subseteq Cl(A \cap T)$. I tried assuming not, and I arrive at the statement $\displaystyle t \in A^c$ and I can't see why that would be a contradiction.

    Does anyone know of a good proof of this? Also, is it necessary that the T be open?
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  2. #2
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    First if I am correct is assuming that $\displaystyle Cl$ is the closure operator, then I am puzzled by your confusion.
    Here are some facts.
    $\displaystyle \left[ {\forall A\,,\,A \subseteq Cl(A)} \right]$, and $\displaystyle Cl(A \cap B) \subseteq Cl(A) \cap Cl(B)$.

    Because A is dense in X you also have $\displaystyle Cl(A) = X$.

    Put all that together.
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  3. #3
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    Plato, you proved the opposite inclusion. You proved that:

    $\displaystyle Cl(A \cap T) \subseteq Cl(T)$

    I want to show that $\displaystyle T \subseteq Cl(A \cap T)$ if T is open.

    As an example if $\displaystyle A = \mathbb{Q}$ and $\displaystyle T = (0,1)$ t hen $\displaystyle (0, 1) \subseteq Cl(\mathbb{Q} \cap (0,1)) = [0, 1]$
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  4. #4
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    Quote Originally Posted by mpetnuch View Post
    Plato, you proved the opposite inclusion.
    No I did not. If T is open then the statement follows as a matter of definitions. If there is a t which is not in $\displaystyle Cl(A \cap T)$ then realize that t is a limit point of A. But some open set O contains t and is a subset of T. Think about that contradiction.

    You asked about T being open. Let $\displaystyle T = \left\{ {\sqrt 2 } \right\}\quad \& \quad A = Q$. It is clear that the statement is false using this example.
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  5. #5
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    Quote Originally Posted by Plato View Post
    No I did not. If T is open then the statement follows as a matter of definitions. If there is a t which is not in $\displaystyle Cl(A \cap T)$ then realize that t is a limit point of A. But some open set O contains t and is a subset of T. Think about that contradiction.
    I am sorry, but I still don't see it. Since $\displaystyle A$ is dense means that $\displaystyle A$ intersects every non-empty open set. In particular $\displaystyle A \cap T \ne \emptyset$. I see this is the reason why it fails if $\displaystyle T$ is not open.

    Also since $\displaystyle A$ is dense means that every point is a limit point of $\displaystyle A$, so in particular $\displaystyle t$ is a limit point of $\displaystyle A \forall t \in T$. And since $\displaystyle T$ is open, we can find an open neighborhood of $\displaystyle T$, $\displaystyle U$, such that $\displaystyle t \in U \subseteq T$.

    Now, you asked me to assume that $\displaystyle t \notin Cl(A \cap T)$ and consider that open set $\displaystyle U$ defined above. Since $\displaystyle t$ is a limit point of $\displaystyle A$ implies that $\displaystyle A \cap U\backslash\{t\}\ne \emptyset$. But I still don't see the contradiction.
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  6. #6
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    Because T is open there is an open set containing t which is a subset of T. Any open containing t must contain must contain a point of A distinct from t. Hence the second point must be in $\displaystyle (A \cap T)$.
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