Topology - Projections I'm pretty sure I have this right, but want to double check and make sure.
Let $X_1$ = $X_2$ = $\mathbb{R}$ and let $p_1: X_1 \times X_2 \rightarrow X_1$ and $p_2: X_1 \times X_2 \rightarrow X_2$ be the projections.  Let $A = {(x,y): 1 \le x \le 2, 3 \le y \le 3x}$.  Find $p_1(A)$ and $p_2(A)$.  
Answer?   
$p_1(A)= [1,2]$ 
$p_2(A)= [3,6]$

Your solution is correct.
Since $A=\{(x,y)\mid 1\le x\le2, 3\le y \le3x \}$, its projection $p_1(A)$ is a subset of $[1,2]$. On the other hand, for each $x\in[1,2]$ there is a $y\in[3,3x]$, so there is a point in $A$ which projects to $x$, thus $p_1(A)=[1,2]$
Since $3\le y\le3x\le6$, we have $p_2(A)\subseteq[3,6]$. And if $3\le y\le6$, then for $x=y/3$ the point $(x,y)\in A$ and projects to $y$, so $p_2(A)=[3,6]$
