Question about product of compact sets I am reading the proof of the fact that the cartesion product of two compact sets is compact. But I ran into the following question.

Let $A$ and $B$ be compact sets in $\mathbb{R}^d$ and
$\{O_\lambda\}_{\lambda\in\Lambda}$ be an open cover of $A \times B$.
For each $(a,b) \in A \times B$, we can choose some $\lambda =
 \lambda(a,b)$ such that $(a,b) \in O_{\lambda(a,b)}$. By construction,
$O_{\lambda(a,b)}$ is open, hence the point $(a,b)$ is contained in
some open box $X \subset O_{\lambda(a,b)}$ where $X = U_{(a,b)} \times
 V_{(a,b)}$, where $U_{(a,b)} \subset A$ and $V_{(a,b)} \subset B$.

Question: I am little confused. How do we know that $U_{(a,b)} \subset A$ and $V_{(a,b)} \subset B$?
I am not asking about the whole proof. Just want to clarify that moment. Please do not duplicate it.
 A: $X$ is a box.  A box is a product of open sets in $A$ and $B$ respectively. The definition of being open in the product implies the existence of just such a box.  Note the product topology is the box topology in the case of a finite product.
That's we're checking for compactness in the product/ box topology.   That's the topology generated by boxes (products of open sets in $A$ and $B$).
Edit:   The product topology on $\Bbb R^{2d}=\Bbb R^d×\Bbb R^d$ is nothing but the standard Euclidean topology. They coincide.  I know I said "topology" again,  and you don't want that,  but my point is just that open sets in $\Bbb R^{2d}$ are all unions of cross products $U×V$, where $U,V\subset \Bbb R^d$.  I'm afraid though that to get $U\subset A,V\subset B$, you may have to consider $A,B$ as subspaces (and here comes that word again).  See compactness is a topological notion.   But relax, you're already familiar with the one on $\Bbb R^n$.  In fact that's probably the prototype.
A: Ok (too long for a comment, but I read your definition of a neighborhood) then do a picture for the plane first as it is easier to visualize (so $d=1$). A neighborhood of a point $p=(a,b)$ is a circle (or rather, an open disk) centered at $p.$ You can inscribe a square in this circle, with horizontal and vertical sides. This square is the product $I\times J$ of two open intervals $I$ and $J$ (each a subset of the respective factor), think $I$ is like the set $U_{(a,b)}$, and $J$ is like the set $V_{(a,b)}$ that the author uses. Except, in the "relative topology" (even if you may not have discussed topology), these neighborhoods you work with are intersected with the sets under consideration, e.g. each $O_\lambda$ is the intersection with $A\times B$ of some set $M_\lambda$ open in the product (or real lines), while $U_{(a,b)}$ and $V_{(a,b)}$ are the intersections with $A$ and $B$ respectively or some open sets $P$ and $Q$ in the respective factor.
