O produto de First Countable por Fréchet é sempre Fréchet?
Jan 15, 2015 20:25:06 GMT -3
Felipe Diniz likes this
Post by Vinicius Rodrigues on Jan 15, 2015 20:25:06 GMT -3
Postei essa questão no Math Stack. Vou postar aqui também.
math.stackexchange.com/questions/1106034/product-of-a-first-countable-space-by-a-fr%C3%A9chet-space
Referência: www.sciencedirect.com/science/article/pii/016686419500034E
math.stackexchange.com/questions/1106034/product-of-a-first-countable-space-by-a-fr%C3%A9chet-space
It's easy to see that the product of two First Countable spaces is First Countable, and it's easy to show that every First Countable space is a Fréchet Space (i.e, if $A \subset X$ and $p \in \bar A$ then there exists a sequence $s_n$ of elements of $A$ such that $s_n \rightarrow p$).
On Alan Dow's "More set-theory for topologists" I saw that there is an nice example of two Fréchet spaces such that their product is not Fréchet. But I was wondering, what if one of them is also First Countable? Will the product be a Fréchet Space?
On Alan Dow's "More set-theory for topologists" I saw that there is an nice example of two Fréchet spaces such that their product is not Fréchet. But I was wondering, what if one of them is also First Countable? Will the product be a Fréchet Space?
Referência: www.sciencedirect.com/science/article/pii/016686419500034E