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TOPOLOGY: PROBLEM SET 4 13th Apr, 2016 Definition (Separation Axioms). Let (X, τ ) be a topological space. We will will say that (X, τ ) is a T# space if (X, τ ) satisfies the property below for all admissible choices of a, b ∈ X and all admissible choices of A, B ∈ τ . We use “T” as an abbreviation for the German word Trennungsaxiom which literally means separation axiom. T0 or Kolmogorov: If a, b ∈ X, a 6= b, there exists an open set U ∈ τ such that either a ∈ U and b ∈ / U , or b ∈ U and a ∈ / U. T1 or Fréchet: If a, b ∈ X, a 6= b, there exist open sets U, V ∈ τ such that a ∈ U , a ∈ / V, b ∈ V , and b ∈ / U. T2 or Hausdorff: If a, b ∈ X, a 6= b, there exist disjoint open sets U and V such that a ∈ U and b ∈ V . T3 : If A is a closed set and b is a point not in A, there exist disjoint open sets U and V such that A ⊆ U and b ∈ V . T4 : If A and B are disjoint closed sets in X, there exist disjoint open sets U and V such that A ⊆ U and B ⊆ V . Regular: We will say that a topological space is regular if the topological space is T3 and T1 . Normal: We will say that a topological space is normal if the topological space is T4 and T1 . Exercise 1. Prove that a Regular space is Hausdorff and that a Normal space is Hausdorff. Date: 13th Apr, 2016. Exercise 2. Determine if the separation axioms are Topological properties, Strong Topological properties or neither. Exercise 3. Let (X, τ ) be a T0 topological space. Prove that there do not exist a finite set of points p1 , p2 , . . ., pn of X such that for each k < n, pk+1 is a limit point of pk and p1 is a limit point of pn . Exercise 4. Prove that if a topological space is Hausdorff, then singletons are closed. Exercise 5. Let (X, τ ) be a Hausdorff topological space. If X is finite, show that τ is the discrete topology. Exercise 6. Prove that the separation axioms T0 , T1 , T2 , and T3 are all hereditary. Find counter-examples that show that the T4 is not hereditary. Exercise 7. Show that pseudometric spaces are T3 and T4 , but not guaranteed to be T0 , T1 , or T2 . Exercise 8. Prove that if a pseudometric space is Hausdorff, then it is a metric space. Exercise 9. Show that any set with more than one element endowed with the trivial topology is not T0 . Exercise 10. Prove that the Zariski topology on a commutative ring is T0 but not T1 . Exercise 11. Show that the cofinite topology on an infinite set is T1 but not T2 . Exercise 12. Show that the product of Hausdorff spaces is Hausdorff. Prove that X Hausdorff topological space iff 4 = {(x, x) : x ∈ X} is closed in X ×X.