Recently I was solving some problems and I ended up with a certain problem that I cant yet understand because of the sets that it proposes. The problem goes like this.
Let be $A\subset \Bbb R^p$ An open set, and $B\subset \Bbb R^n$. Lets define the following sets
a) A+B={ x+y$\in \Bbb R^p$ | $x\in A$ and $y\in B$}
b) If n=1, AB={ xy$\in R$ | $x\in A$ and $y\in B$}
c) If $A\subset \Bbb R$, E={ (x,y)$\in \Bbb R^2$ | x$\in A$}
Determine which of the previuos sets are open sets.
And that is the problem, but my concern is that, can it be possible to define such kinds of sets?, becuase for example in a) for me it seems that the elements of the set are defined by the addition of elements from diferent vectorial spaces, and I dont understand if it is posible or not. I would appreciate any explanations about this in order for me to undestand and solve this problem.