Are there affine spaces that contain subsets that aren't closed to affine combinations of three points?
This is a surprising question.
I think that exists that kind of affine spaces,but I don't know what space should be considered as an example.
Please,could you help me?
A simple affine space would be the (x,y) plane, where we use the vector space $\mathbb{R}^2$ as the vector space $V$ of the definition. For two points $a,b$ in the (x,y) plane we can define $u(a,b)=b-a,$ for example $u((1,2),(2,5))=(2,5)-(1,2)=(1,3).$ The origin $(0,0)$ of the (x,y) plane can then be the point $O$ of the second part of the definition, since under the mapping $u$ a random point $a=(x,y)$ of the plane maps to $u((0,0),(x,y))=(x,y)$ which shows this maps the (x,y) plane as an affine space by a bijection to the vector space $V=\mathbb{R}^2.$
Now for a subset $B$ which is not closed under affine combinations, take the three-element set consisting of the vertices $P,Q,R$ of a nondegenerate triangle. Then the affine combination $(1/3)P+(1/3)Q+(1/3)R$ is the centroid of the triangle, and is not one of the vertices, so is not in $B.$