Usage of commas in sets as "such that" I can't find this anywhere, but a particular lecturer that I have seems to be using commas to mean "such that" in sets. I'm seeking clarification on whether this really is the case or if I'm just misinterpreting.
Example:
$$\{U_1 \times U_2, U_1 \textit{open in }X_1, U_2 \textit{open in }X_2\}.$$
Is this the same as
$$\{U_1 \times U_2| U_1 \textit{open in }X_1, U_2 \textit{open in }X_2\}?$$
 A: It is not wrong to use "$,$" for "such that" in this context (however, it is not usual). In fact, from the logical point of view, we can use any symbol to denote anything (after all, truth is invariant under changes of notation). The point is: the notation's meaning have to be at least (i) clear for the reader and (ii) unambiguous.
In the first set in your post, the symbol "$,$" has two different meanings (as explained in the @par 's comment). So, I'd say that the comma's use is being inappropriate. But, of course, for the experienced reader this use will not cause problems.

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. (Alfred North Whitehead) - From Tao's blog.

A: Your interpretation is correct. The usual (and better) notation is $$\{U_1 \times U_2:\, U_1 \text{ open in }X_1 \text{ and } U_2 \text{ open in }X_2\}.$$
A: No one can say that it's wrong to use a comma like that since we might just not have seen the convention your teacher is using, but I've never seen a convention where you can use a comma like that. Either "$:$" or "$|$" would be fine in place of the comma.
