 # Topology Problems

## Question

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?

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