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Math Help - What is the definition of a convex subset of a topological space

  1. #1
    CSM
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    What is the definition of a convex subset of a topological space

    I'm trying to show that the Least Core of \Pi is a convex set whenever \Pi is a convex and closed set in a topological vector space.
    The definition of the Least Core is:
    LC(\Pi,F):=\{x \in\Pi |  \bigvee_{j\in M}F_j(x)\leq\bigvee_{j\in M}F_j (y) \forall y\in  \Pi\} Where \bigvee is the maximum operator and M is the index set of all the funtions on \Pi


    But what is the definition of a convex subset of a topological space that I should use?
    Can I just use this notion:
    Let V be a vector space (over R or C). A subset S of V
    is convex if for all points x,y in S, the line segment
    \{\alpha x + (1-\alpha) y \mid \alpha\in(0,1)\} is also in S.


    Btw this is no homework, so I don't know exactly what definition to use.
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  2. #2
    CSM
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    Solved it.


    Let $S$ be the Least Core of $(\Pi,F)$. Let $x_1,x_2 \in S$. Show for an arbitrary $\alpha \in (0,1) \,\alpha x_1 + (1-\alpha) x_2$ is contained in $S$.
    $x_1,x_2\in S$ so $\bigvee_{j\in M}F_j(x_1)\leq\bigvee_{j\in M}F_j (y)\,, \forall y\in \Pi$ and $\bigvee_{j\in M}F_j(x_2)\leq\bigvee_{j\in M}F_j (y)\,, \forall y\in \Pi$
    Because the $F_j$'s are convex we know$F_j(\alpha x_1+(1-\alpha )x_2)\leq \alpha F_j(x_1) + (1-\alpha)F_j(x_2)$. Without loss of generality we may assume $\bigvee F_j(x_2)\geq \bigvee F_j(x_1)$, so: $\bigvee F_j(\alpha x_1+(1-\alpha )x_2)\leq \bigvee F_j(x_2)$ and because $x_2$ lies in the least core, so does $\alpha x_1+(1-\alpha )x_2$\\
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  3. #3
    CSM
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    The definition of the Least Core is:
    LC(\Pi,F):=\{x \in\Pi |  \bigvee_{j\in M}F_j(x)\leq\bigvee_{j\in M}F_j (y) \forall y\in  \Pi\} Where \bigvee is the maximum operator and M is the index set of all real funtions F on \Pi


    How to proof that LC is closed? I tried proving it using a sequence in LC but it made no sense
    Last edited by CSM; May 18th 2011 at 05:01 AM.
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