Consider and the topology on whose basic neighborhoods for are the intervals .
Then, is normal however it is not normal.
Regards.
Fernando Revilla
Can somebody help me solve this problem?
Let be regular for each . Then the product space is regular.
My biggest problem is that I don't know how closed sets look like in a product space. Is that an infinite product of closed sets? is closed for all . Or need only a few of 's to be closed? What about the rest? Is it empty?
Consider and the topology on whose basic neighborhoods for are the intervals .
Then, is normal however it is not normal.
Regards.
Fernando Revilla
Suppose that is a non-empty class of regular spaces and let have the product topology. Recall that it suffices to prove that given and a neighborhood of it there exists a neighborhood of such that .
So, let be any basic neighborhood of with the finitely many indices for which . Then, for each we may find some neighborhood of for which and . Clearly where is a neighborhood of and . By prior discussion the conclusion follow.s