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Is it true that all convex sets are connected? This seems intuitively true, but I would like to know if it is, then why? And if not, why not? In other words, is showing that a set is convex sufficient to show that it is connected?

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Hint: use path connectedness. –  Emilio Ferrucci Feb 4 '12 at 0:04
    
Is this a pure topological question, or a topological vector space question? –  Tim Feb 4 '12 at 0:08
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Let $C$ be a convex set. Suppose is not connected then there are to disjoint non empty open sets $U,V$ so that $U\cup V=C$. Pick points $x\in U$ and $y\in V$ since $C$ is convex the line segment between them is contained in $C$, i.e., the function $f: [0,1]\to C$ defined $f(t)=tx+(1-t)y$. Note that $f^{-1}(U)$ and $f^{-1}(V)$ would show that $[0,1]$ is not connected a contradiction.

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Consider the set X={a,b} endowed with discrete topology and a order relation. It is convex but not connected...

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You are misunderstanding the notion of convexity used in the question. –  Mariano Suárez-Alvarez Jul 9 '12 at 4:29
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