NOTE: I will use the symbol $"Y<X"$ to denote $Y$ is an affine subspace of $X$
I have this proposition:
Let $X$ be an affine space. If $M<X$ and $N<X$ such that $M \cup N < X$ $\Longrightarrow$ $M \subseteq N$ or $N \subseteq M$
I think this implication is not true, but I'm not sure... Using the hypothesis, I have concluded that $M \cup N = M + N $ and my intuition says the following:
In the hypothetical case that an affine space with two different elements exists (I'm not sure about that). In other words, $X = \{a,b\}$ where $a \neq b$, then the two unipuntual sets $M =\{a\}$ and $N =\{b\}$ are affine subspaces of $X$ and $M \cup N = X < X$, but $M\nsubseteq N$ and $N \nsubseteq M$ .
So... Is there any affine space with these characteristics?
Let $X$ be the two-dimensional subspace over the field of two elements, $X=\{(0,0),(0,1),(1,0),(1,1)\}$. Let $M=\{(0,0),(0,1)\}$, $N=\{(1,0),(1,1)\}$. Then $M$ and $N$ are affine subspaces of $X$, their union is $X$, but neither one contains the other.