Product of induced topologies and induced product topology coincide?

Question

Let $X$ and $Y$ be two topological spaces, $A\subset X, B\subset Y$. I have to prove that the following two topologies on $A\times B$ coincide:

The product of the topology of $A$ (induced from $X$) and that of $B$ (induced from $Y$),

and the topology induced on $A\times B$ from the product topology on $X\times Y$.

Intuitively, I know this is correct. But how do I prove this mathematically correct?

Thank you!

http://math.stackexchange.com/a/1548818/4280 proves a more general statement. — Henno Brandsma, Dec 22 '15 at 12:05
@HennoBrandsma I don't understand which part of that proof is answer to my question.. — jbuser430, Dec 22 '15 at 12:26
At the end (products and subspaces), as an application of the transitive law of initial topologies. — Henno Brandsma, Dec 22 '15 at 12:28

score 2 · Answer 1 · answered Dec 22 '15 at 12:23

$(U \cap A) \times (V \cap B) = (U \times V) \cap (A \times B)$ (where $U$ open in $X$, $V$ open in $Y$) is a key identity here. The right hand side describes a base for the subspace topology for $A \times B$, the left hand side a base for the product topology of the subspace topologies. So the topologies share a base and are identical.

Product of induced topologies and induced product topology coincide?

1 Answers1