Download TOPOLOGY: PROBLEM SET 4 Definition (Separation Axioms). Let

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.