question about product toplogy?

3 ⤊

question about product toplogy?

In product toplogy, the sets of the form U_1 x U_2 x ... x U_n are open, where U_i are open . Are there any other sets which are open?
Thanks

2007-01-27 18:19:45 · 2 answers · asked by Theta40 7 in Science & Mathematics ➔ Mathematics

2 answers

Sure. The union of two open sets is always open. Consider a product topology on A x S. Let B and C be open subsets of A, and T and U be open subsets of S. Then B x T and C x U are both open, and so is their union. If B is a strict superset of C and U is a strict superset of T, this is a counterexample to the claim that the only open sets are of the form already given.

2007-01-27 18:40:21 · answer #1 · answered by Curt Monash 7 · 2⤊ 0⤋

Also their covers and sub-covers if happen to be any.

2007-01-28 02:28:11 · answer #2 · answered by Anonymous · 1⤊ 1⤋