1. Math
  2. Advanced Math
  3. 14 definition a topological space x t is called perfectly...

Question: 14 definition a topological space x t is called perfectly...

Question details

14. Definition: A topological space (X, T) is called perfectly normal if and only if X is normal and every closed subset of X is a Gs-set. A topological space (X, T) is called To-space if and only if X is T and perfectly normal. From the definition, it is clear that a normal space X is perfectly normal if and only if any open subset is an F,-set. Obviously, any discrete space and any T4 countable space are perfectly normal. (R, U) is To. In fact. any metrizable space is T6- (a) Prove that any normal regular second countable space is perfectly normal (b) Prove that the Sorgenfrey line is To Hint: You may use the following fact about (R, U), the usual topology: any open set U EU is a union of countably many open bounded intervals.

Solution by an expert tutor
Blurred Solution
This question has been solved
Subscribe to see this solution