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I was trying to visualize all affine sets in $\mathbb{R}^2$.

I started with single points. All points are affine sets of cardinality one.

Now I thought of adding another point to the context. The set of two points is not affine. But the affine hull is a line through these points which is an affine set.

Now, I thought of adding a point which is not in the line . The union of line and the new point is not affine. The affine hull cor this union set is $\mathbb{R}^2$.

So the affine sets are {singleton points ,lines, $\mathbb{R}^2$}

Is this right?

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  • $\begingroup$ Yes, this is right. $\endgroup$ – Theo Bendit Oct 13 '18 at 15:23
  • $\begingroup$ Another way to look at this is to remember affine spaces are just translates of vector spaces (and conversely), and vector subpaces are caracterised by their dimension. So yes, inside a plane (of dim 2), there are, the singleton $0$ (of dim 0), all lines through the origine (of dim 1), and the plane itself. $\endgroup$ – Drike Oct 13 '18 at 16:08

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