Hello: I have what may be a silly question because did not find it in the books I looked for it and is this: Is the product of closed sets closed in the product topology?
The answer seems to me to be "yes".
Hello: I have what may be a silly question because did not find it in the books I looked for it and is this: Is the product of closed sets closed in the product topology?
The answer seems to me to be "yes".
Yes, the product of two closed sets is closed in the product topology, just as the product of two open sets is open.
But be careful! just as not every set in X x Y is the product of sets in x and y, so closed (or open) sets in X x Y may not be the product of closed (or open) sets in X and Y.
For example, the singleton set {(x, y)) in X x Y is the product of the two singleton sets {x} and {y} but {(x,y), (x, z)} in X x Y (so both y and z are in Y) is NOT the product of two sets in X and Y.
Theorem: let be a non-empty class of topological spaces and let be under the product topology. Then, given any class of subsets such that we have that
Proof: Let . Let be any neighborhood of . Clearly we may find a basic open set (in the defining open base) such that . Since is an adherent point of we may pick some . Clearly then . Since was arbitrary it follows that is an adherent point for and thus
Conversely, let . Let be arbitrary and consider any neighborhood of . Since is open in the product topology we see that it contains a point . It follows then that . In other words, is an adherent point for and thus . It follows that .
The conclusion follows.
There are two nice corollaries
Corollary:Let be a collection of closed sets such that . Then from where it follows that is closed.
Corollary: Let be a collection of dense subsets of . Then, . From where it follows that is dense in