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From Theory of Convex Structures by M. L. J. Van De Vel, on a set $X$, a topology and a convexity structure are said to be compatible, if the convexity structure is generated by the closed sets. The condition is equivalent to that all the polytopes (the convex hulls of finite sets are called polytopes) are closed. Then the set $X$ together with the compatible topology and convexity structure is called a topological convex space (TCS).

In the next page, it says that the a topological vector space (TVS) over $\mathbb{R}$, i.e. a real vector space with a Hausdorff topology such that addition and scalar multiplication are both continuous, is a TCS with the standard convexity structure, because every polytope is compact.

My question comes from my understanding of the above two paragraphs. From the definition of a TCS, a closed set is convex. But in $\mathbb{R}^n$, as an example of a TVS over $\mathbb{R}$, there are closed sets that are not convex, such as a half of a sphere. So I wonder what mistakes I have made?

Thanks and regards!

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"From the definition of a TCS, a closed set is convex." This doesn't follow from the condition that all polytopes are closed. –  Qiaochu Yuan Jan 5 '13 at 4:20
    
The first link says that the convexity structure is generated by closed sets, so I think all the closed sets have to be convex to be a generator? –  Tim Jan 5 '13 at 4:25
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@Tim It is generated by certain closed sets, not all closed sets. –  Alex Becker Jan 5 '13 at 4:36
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(repeating Alex Becker's point) You misquote the definition in your question: van de Vel writes "... if the convexity structure is generated by closed sets." and you write "... if the convexity structure is generated by the closed sets." This one word makes a world of difference and is likely the source of the confusion. –  Martin Jan 5 '13 at 4:38

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