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Proof: Topology of subsets on a Cartesian product 
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#1
Jan1012, 08:22 PM

P: 1

The problem statement, all variables and given/known data
Let T_{x} and T_{y} be topologies on X and Y, respectively. Is T = { A × B : A[itex]\in[/itex]T_{x}, B[itex]\in[/itex]T_{y} } a topology on X × Y? The attempt at a solution I know that in order to prove T is a topology on X × Y I need to prove: i. (∅, ∅)[itex]\in[/itex]T and (X × Y)[itex]\in[/itex]T ii. T is closed under finite intersections iii. T is closed under arbitrary unions In order to prove (i) I would have to prove that ∅[itex]\in[/itex]A and ∅[itex]\in[/itex]B. I think this is true because the empty set is in all sets. I'm not sure how to approach proving that X[itex]\in[/itex]A as even though A[itex]\in[/itex]T_{x}, this implies that A[itex]\in[/itex]X or A is X. I'm not sure how continue from here. Same with Y[itex]\in[/itex]B. For ii. I think that since T_{x} and T_{y} are topologies themselves, they are closed under finite intersections, and since A[itex]\in[/itex]T_{x} and B[itex]\in[/itex]T_{y} then A and B are also closed under finite intersections, thus T is closed under finite intersections. I have to go more into detail with this but I just want to make sure if this is the right idea. I think iii. could also be proved with a similar argument to the one used to prove ii. 


#2
Jan1012, 11:25 PM

P: 47

For (i), your topology T is the set of all open sets A x B such that A is an element of T_x, and B is an element of T_y. X is an element of T_x, as it is required to be one by the same rules we a re trying to prove, as well as the empty set, and vice versa for Y being an element on T_y. Use this fact to show that X x Y is in your topology T.



#3
Jan1112, 12:34 AM

Emeritus
Sci Advisor
PF Gold
P: 9,530

##A\in T_x## doesn't imply what you say it implies. It just means that A is an open subset of X. Note that the definition of T says that T consists of of all cartesian products of two open subsets of X and Y, such that the first set is a subset of X and the second a subset of Y. To check if ##\emptyset\in T##, you should ask yourself if ∅ can be expressed as a cartesian product at all. X×Y is obviously a cartesian product, so to prove that X×Y is in T, you only have to prove...what? 


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