Let $I=[0,1]$. Compare the product topology on $I\times I$ , the dictionary order topology on $I\times I$ , and the topology $I\times I$ inherits as a subspace of $\mathbb{R}×\mathbb{R}$ in the dictionary order topology.
First of all, I know that the product topology on $X\times Y$, being $X$ and $Y$ topologies, is generated by the collection of sets of the form $U\times V$ where $U$ is open in $X$ and $V$ is open in $Y$. Therefore, $I\times I$ will be generated by such collection. But what are opens in $I$? I must assume some kind of standard topology? Should it be the one generated by opens in the form $(a,b)$? I think no, because there's no basis element of this form that contains $0,\frac{1}{2}$ for example.
Now, for the dictionary order topology, I must assume basis elements of the form:
$$(a,b), [a_0,b), (a,b_0]$$
where $a_0 = 0$ and $b_0 = 1$
right?
And what about the third topology?
I know that comparing them means finding if some topology is contained in another, but how to compare such strange topologies?
Best Answer
Hint: The basis elements of each topology are:
The subspace topology inherited from the standard topology on $\mathbb R^2$: $$ ((x_1, x_2) \cap [0, 1]) \times ((y_1, y_2) \cap [0, 1]) $$
The dictionary order topology on $I \times I$ has three types of basis elements: $$ (x_1 \times y_1, x_2 \times y_2), [0 \times 0, x_2 \times y_2), (x_1 \times y_1, 1 \times 1] $$
The subspace topology inherited from the product topology on $\mathbb R_d \times \mathbb R$, where $\mathbb R_d$ is the reals with the discrete topology: $$ \{x\} \times ((y_1, y_2) \cap [0, 1]) $$