Question 1.
Recall that (X,T) is a T₄ topological space if and only if given disjoint closed subsets, K and L of X, there are disjoint open subsets, U and V, of X with K ⊆ U and L ⊆ V
Prove that (X, T) is a T₄ topological space if and only if given any closed subset F of X and any open subset, H of X with F C H, there is an open subset, G, of X such that F ⊆ G ⊆ Ḡ ⊆ H
Question 2.
Prove that every metric space is normal.
Question 3.
Prove that every closed subspace of a normal space is normal.
Question 4.
Let X be a set. Show that T : = { A ⊆ X | A = ∅ or X \ A is finite } is a topology on X. This is the finite complement topology on X. When is it metrisable?
Question
Solution Preview
These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.
By purchasing this solution you'll be able to access the following files:
Solution.pdf.
Return to homework library