Here's the exercise:
Let $A$ and $B$ be separated subsets of some $\mathbb{R}^k$, suppose $a \in A$, $b \in B$, and define: $$p(t) = (1-t)a + tb$$ for $t \in \mathbb{R}^1$. Put $A_0 = p^{-1} (A)$, $B_0 = p^{-1} (B)$ [Thus $t \in A_0 \iff p(t) \in A$]
My problem is that I don't get just what is $p(t)$. Is it a mapping? If so, then what is $p(A)$? Is it a set?
I'm sorry if it sounds stupid or too elementary but I simply don't understand what is this $p(t)$ Rudin just defined. It resembles a property of convexity - but it accepts any real $t$, not just a real $t$ such that $0<t<1$.
I think this has something to do with convexity in finite dimensions, but I'm not sure.
$p: \mathbb{R} \to \mathbb{R^n}: t \mapsto p(t)$ is a map.
Geometrically, the graph of the function is the line between the points a and b.
Btw, I dislike Rudin's approach to show that convex sets are connected. It's easier and more general to show that pathconnected sets are connected and then it's trivial to see that convex sets are connected (using the map that Rudin provides).