Franklin's Notes

Affine space

An affine space is a kind of generalization of the concept of a vector space that avoids making an arbitrary choice of origin. Every affine space consists of a set of points $X$ and has an underlying vector space $T$ such that there is a difference function with the property that and that the partially evaluated map $d_x:X\to T$ sending $y\mapsto d(x,y)$ is a bijection between $X$ and $T$. This function assigns a vector value to the "separation" between any two points of the space $X$. Notice that every vector space $X$ has an induced affine structure given by the map

By letting $x=y=z$ in the additivity equation for $d$, we obtain the equality $2d(x,x)=d(x,x)$, or $d(x,x)=\vec{0}$. Letting $z=x$, we obtain $d(x,y)+d(y,x)=0$, or $d(y,x)=-d(x,y)$. These equalities express the fact that "every point has zero displacement from itself", and that "the displacement from $x$ to $y$ is the opposite of the displacement from $y$ to $x$".

A topology on an affine space over $\mathbb R^n$ is often defined by making use of charts , by fixing a chart $C_a$ and setting $S\subset X$ open in $X$ iff $C_a(S)$ is open in $\mathbb R^n$. It can be proven that this induced topology is independent of the choice of chart $C_a$, so it can simply be called the "usual topology" on the affine space $X$.




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