Questions: Regularity and T₃ Spaces

Question 1

A topological space X is called regular if:

• A. Any two distinct points can be separated by disjoint open sets
• B. Any point and any closed set not containing it can be separated by disjoint open sets
• C. Any two disjoint closed sets can be separated by disjoint open sets
• D. Every open cover of X has a finite subcover

Answer: (B)

Regularity separates a point from a closed set. Option A is the Hausdorff (T₂) axiom — separating two points. Option C is normality (T₄) — separating two closed sets. Option D is compactness. The hierarchy T₂ < T₃ < T₄ represents successively stronger separation requirements: from two points, to a point and a closed set, to two closed sets.

Question 2

A topological space X is Hausdorff (T₂) but fails to be regular. Which configuration would witness this failure?

• A. Two distinct points that cannot be separated by open sets
• B. A point x and a closed set F with x ∉ F such that every open set containing x intersects every open set containing F
• C. An open set whose complement is not closed
• D. A continuous bijection from X to another space that is not a homeomorphism

Answer: (B)

Regularity fails when a point and a disjoint closed set cannot be placed in disjoint open neighborhoods. Option B directly describes this: x and F cannot be separated, even though they are topologically distinct objects (x is not in F). A space can be T₂ (separating any two points) while failing this stronger condition — there exist pathological spaces that separate points but cannot separate a point from a closed set.

Question 3

Every metric space is regular: given a point x and a closed set F with x ∉ F, open balls of radius r < d(x, F)/2 around x and around points of F provide the required disjoint open separation.

• T. True
• F. False

Answer: True

In a metric space, d(x, F) = inf_{y ∈ F} d(x, y) > 0 because F is closed and x ∉ F. Open balls B(x, r) and ∪_{y∈F} B(y, r) for r = d(x,F)/2 are disjoint: any point in both would satisfy d(x, z) < r and d(z, y) < r for some y ∈ F, giving d(x, y) < 2r = d(x, F) — a contradiction. This is why metric spaces are well-behaved topologically: the metric provides the geometric tools to construct separation.

Question 4

A regular topological space automatically satisfies the Hausdorff (T₂) axiom — that any two distinct points can be separated by disjoint open sets.

• T. True
• F. False

Answer: False

Regularity alone does not imply T₂. Consider an indiscrete space with two points: every 'closed set' is either empty or the whole space, so the regularity condition is vacuously satisfied (there's no closed set disjoint from any point that you need to separate). But the two points cannot be separated by open sets. The T₃ axiom adds the T₀ condition (distinct points are topologically distinguishable), and together T₀ + regular does imply T₂. The subscript naming matters: T₃ = regular + T₀, not just regular.

Question 5

What is the key difference between the T₂ (Hausdorff) axiom and the regularity axiom, and why does this make regularity a stronger form of separation?

Model answer: T₂ separates two points: given distinct x, y, there exist disjoint open sets U ∋ x and V ∋ y. Regularity separates a point from a closed set: given x and closed F with x ∉ F, there exist disjoint open sets U ∋ x and V ⊇ F. Separating a point from an entire closed set is a more demanding task than separating two points, because a closed set can be infinite and its points can cluster arbitrarily close to x. Regularity (combined with T₀) implies T₂ — since {y} is closed in a T₁ space, regularity of x from {y} gives the T₂ separation — but T₂ does not in general imply regularity.

The progression T₀ ⊂ T₁ ⊂ T₂ ⊂ T₃ ⊂ T₄ represents a strict hierarchy of separation strength, with each level solving a harder separation problem. Regular + T₀ = T₃ sits above T₂ because it separates points from closed sets rather than just pairs of points.