Given an affine space $X$, we say that $X'\subset X$ is an **affine subspace** or **flat** of $X$ if

1. $d(X'\times X')$ is a vector subspace of the underlying vector space $T$ of $X$, and

2. $X'$ is an affine space with underlying vector space $d(X'\times X')$ and difference function given by the restriction of $d$ to $X'\times X'$

Additionally, given a set $S$, the **affine subspace generated by $S$** is defined to be the "smallest affine subspace containing $S$", or the intersection of all affine subspaces containing $S$. This does in fact define an affine subspace because it can be shown that arbitrary intersections of subspaces of vector spaces are also subspaces, meaning that arbitrary intersections of affine subspaces are also affine subspaces. For instance, three noncollinear points generate a plane in the induced affine space on $\mathbb R^n$.

The **translate** $X'+t$ of an affine subspace $X'\subset X$ by a vector $t\in T$ is said to be the affine subspace ${x+t:x\in X'}$. Additionally, two affine subspaces are said to be **parallel** if $d(X'\times X')=d(X''\times X'')$. It happens that $X',X''\subset X$ are parallel iff $X''=X'+t$ for some vector $t$.

Proposition 1.Two affine subspaces $X',X''\subset X$ are parallel iff there exists $t\in T$ such that $X''=X'+t$.

Proof.First, suppose that $X',X''$ are parallel, so that $d(X'\times X')=d(X''\times X'')$. If we choose $x'\in X'$ and $x''\in X''$ arbitrarily, then we may let $t=d(x',x'')$. Now, if $y''\in X''$ is arbitrary, we have that $d(x'',y'')$ is in $d(X''\times X'')=d(X'\times X')$, so that and the quantity $x'+d(x'',y'')$ is in $X'$, since $d(x'',y'')\in d(X'\times X')$. Conversely, given arbitrary $y'\in X'$, we have $d(x',y')$ is in $d(X'\times X')=d(X''\times X'')$, so that and $x''+d(x',y')$ is in $X''$ since $d(x',y')\in d(X''\times X'')$. Hence, every $y''\in X''$ equals $y'+t$ for some $y'\in X'$ and every $y'\in X'$ equals $y''-t$ for some $y''\in X''$, meaning that $X''=X'+t$.Now suppose that $X''=X'+t$. Then, for any $x'',y''\in X''$, we have that meaning that $d(X''\times Y'')\subset d(X'\times Y')$ and similarly we may show that so that as desired. $\blacksquare$

Conjecture 1.If $X',X''$ are two parallel affine subspaces of $X$, then the set $T'$ of vectors $t$ such that $X''=X'+t$ is an affine subspace of the induced affine space on $T$.

Exercise 1. (Dodson & Poston, II.1.2a)Find a subset $S\subset \mathbb R$ such that any vector $v\in\mathbb R$ occurs as $u-w$ for some $u,w\in S$, but such that $S$ isnota flat of $X$.

Solution.Consider the set $S=\mathbb R^{\ge 0}$ of nonnegative reals. Any real number $v\in\mathbb R$ is a difference of two nonnegative reals, but the set of nonnegative reals is not an affine subspace of $\mathbb R$, in particular because the partially-evaluated difference functions $d_x$ are not bijective on $\mathbb R^{\ge 0}$. For instance, if we restrict $d$ to $\mathbb R^{\ge 0}$, then $d_1$ is not surjective, because there is no $x\in \mathbb R^{\ge 0}$ such that $d_1(x)=-2$ (because $1-2=-1$ is not in $\mathbb R^{\ge 0}$).