I am reading Topology by James Munkres and he defines the dictionary order relation as:
Definition Suppose that $A$ and $B$ are both sets with order relations $<_A$ and $<_B$ respectively. Define an order $<$ on $A\times B$ by defining
$$a_1 \times b_1 < a_2 \times b_2$$
if $a_1<_A a_2$, or $a_1=a_2$ and $b_1<_B b_2$. It is called the dictionary order relation on $A\times B$.
I think that I intuitively understand the relation as working as indexing words in the dictionary. The problem is that I do not understand how is it that the order relation of the following cases is defined in the previous definition:
- How is it in the definition that $a_1 \times b_1 < a_2 \times b_1$ ?
- How is it in the definition that $a_1 \times b_3 < a_3 \times b_1$ ?
I kind of intuitively feel that it could be deduced from the definition but I do not see how. I want to understand it so I make sure I do understand the concept and the definition.