Given two topological spaces X = {a,b} and Y = {1,2,3} find an open set in XxY that is not a product of an open set in X and an open set in Y

Question

Let $X = \{a,b\}$ have the topology $T_x =\{\varnothing, X, \{a\}\}$.

Let $Y = \{1,2,3\}$ have the topology $T_y=\{\varnothing, Y, \{1\}, \{3\}, \{1,3\}, \{2,3\}\}$.

Find an open set in $X\times Y$ that is not a product of an open set in $X$ and an open set in $Y$.

This is an exam review, not a homework problem. So if someone could give me an example with a little explanation that would be awesome.

My initial thought was to take the union as follows:

$\{a\}\times\{1,3\}\cup \{a\}\times\{2,3\} = \{a\}\times\{1,2,3\}$

But I quickly realized, the union of open sets will always be the product of an open set from $X$ and an open set in $Y$. When I turned this worksheet in weeks ago I put the above as my solution, got it wrong, then forgot to ever ask about it.

I'm also thinking this is the same with the intersection of open sets:

$\{a\}\times\{1,3\}\cap \{X\}\times\{2,3\} = \{a\}\times\{3\}$, which is the product of an open set in $X$ and an open set in $Y$.

Does it mean to take the product of non-open sets and create an open set such as:

$\{a\}\times\{1,2\}\cap \{a\}\times\{1,3\}$

Or is there some other way to do this?

Answer 1

I suggest you

1. Write down all products of open sets; there are only 10 of them aside from the empty set, so 11 in total. And these form a basis for the topology on the product.

2. Look for other open sets by taking unions and intersections of the items in the list, and see whether the result is also in the list. As you've already observed (perhaps correctly -- I haven't checked) the list is closed under unions, so all you need to consider are intersections.

This won't take long, and it will be instructive.

Answer 2

But I quickly realized, the union of open sets will always be the product of an open set from X and an open set in Y.

This is where you went wrong. Your first example didn't work because it was of the form $(U \times V_1) \cup (U \times V_2) = U\times (V_1 \cup V_2)$. The $U \subseteq X$ was the same in both products. What if it were different?

Answer 3

HINT: $$\begin{array}{cc|c|} \begin{array}{ccc} \end{array}& \begin{array}{ccc} \end{array}& \begin{array}{cc} X \end{array}\\ &&\begin{array}{ccc} a&b \end{array}\\ \hline \begin{array}{ccc} Y \end{array}& \begin{array}{ccc} 1\\3\\2 \end{array}& \begin{array}{ccc} \bullet&\\ \bullet&\bullet\\ \bullet&\bullet \end{array}\\ \hline \end{array}$$