- #1
redrzewski
- 117
- 0
I'm trying to understand Munkres' proof of the Nagata-Smirnov metrization theorem : A space X is metrizable iff X is regular and has a basis that is countably locally finite.
He takes the union of all the Bn collections of the countably locally finite basis. Roughly, he then constructs a function F: X -> [0,1]^J where J is a subset of ZxB where B is all the collection of all elements in the union of all the Bn's.
Then he concludes that this imbedding implies X is metrizable.
All this I follow. But I thought that if J was uncountable, then [0,1]^J isn't metrizable. And hence his result would only follow if we could show that J is countable.
Am I on the right track? Do I need to show that J is always countable? Or is [0,1]^J with uncountable J metrizable?
thanks
He takes the union of all the Bn collections of the countably locally finite basis. Roughly, he then constructs a function F: X -> [0,1]^J where J is a subset of ZxB where B is all the collection of all elements in the union of all the Bn's.
Then he concludes that this imbedding implies X is metrizable.
All this I follow. But I thought that if J was uncountable, then [0,1]^J isn't metrizable. And hence his result would only follow if we could show that J is countable.
Am I on the right track? Do I need to show that J is always countable? Or is [0,1]^J with uncountable J metrizable?
thanks