6 янв. 2017 г. · The product topology on X×Y is the topology having basis the collection B of all sets of the form U×V, where U is an open set of X and V is an open set of Y.

18 сент. 2018 г. · No, we do not define the product topology to be {U×V:U∈TX,V∈TY} we define it to the the topology with basis {U×V:U∈TX,V∈TY}.

8 июл. 2021 г. · The product topology is the one that satisfies the universal property of products. – Christoph. Commented ...

19 июл. 2014 г. · The product topology is generated by sets of the form ∏n∈NUn where each Un is open in Xn and, for all but finitely many n, we have Un=Xn. In ...

13 окт. 2015 г. · I understand that our product topology for A×B is that a set (u,v) is open in it if and only if u is open in A and v is open in B.

18 янв. 2017 г. · A topology generated by a subbasis is the collection of all unions of finite intersections of subbasis elements.

20 окт. 2021 г. · We know that for topological spaces (X,TX) and (Y,TY) the product topology is the topology on X×Y generated by the basis B={U×V∣U∈TX∧V∈TY}. I ...

24 окт. 2012 г. · Product topology only collects finite intersections of inverse images under projections as open sets. Can someone give more examples outside this property?

13 окт. 2013 г. · The product topology is the one generated by the sets ∏Uα, where Uα⊂Xα is open for all α, and for all but finally many indices Uα=Xα. The box ...

5 мар. 2017 г. · Let ⟨x,y⟩∈U. On base of the definition of "product topology" we conclude that some Vx∈TA and some Wy∈TB exist with:⟨x,y⟩∈Vx×Wy⊆U.