Questions: Regular Spaces (T3 Spaces)

Question 1

A topologist claims: 'This space is Hausdorff, so it must be regular.' Is this correct?

• A. Yes — Hausdorff (T2) is a stronger condition than regular (T3), so T2 always implies T3
• B. No — Hausdorff separates points from points; regularity additionally requires separating points from closed sets; T2 does not imply T3 in general
• C. Yes — the separation axiom hierarchy guarantees stronger axioms imply weaker ones in every space
• D. Only if the space is also compact — compact Hausdorff spaces are always regular

Answer: (B)

Hausdorff and regularity are different separation conditions targeting different objects. T2 separates any two distinct points with disjoint open sets. T3 separates a point from an arbitrary closed set not containing it — a closed set is typically much larger than a single point, so this is a genuinely stronger demand. There exist Hausdorff spaces that are not regular. Option D contains a true fact (compact Hausdorff spaces are indeed normal, hence regular), but it doesn't validate the general claim.

Question 2

What is the key difference between what Hausdorff (T2) and regular (T3) spaces can do?

• A. T2 applies to finite topological spaces; T3 applies to infinite ones
• B. T2 separates any two distinct points with disjoint open sets; T3 separates any point from any closed set not containing it
• C. T2 requires that every closed set is also open; T3 removes this requirement
• D. T2 and T3 are equivalent conditions — the labels are historical accidents

Answer: (B)

This is the definitional core. T2 asks: given two distinct points, can we find disjoint open neighborhoods? T3 asks a harder question: given a point x and a closed set F not containing x, can we find disjoint open sets separating them? The closed set F may contain infinitely many points, so separating a single point from an entire closed structure is a strictly stronger demand.

Question 3

Every T3 space (regular and T1) is automatically Hausdorff (T2).

• T. True
• F. False

Answer: True

In a T3 space, single points are closed (the T1 condition). Regularity then allows separating any point from any closed set not containing it. Since {y} is closed (by T1) and x ∉ {y} for distinct x and y, regularity gives disjoint open sets separating x from {y} — exactly the Hausdorff condition. So T3 implies T2 in the presence of T1, confirming the separation axioms form a genuine hierarchy: T1 ⊂ T2 ⊂ T3 ⊂ T4.

Question 4

A space is normal (T4) if it can separate any point from any closed set not containing it with disjoint open sets.

• T. True
• F. False

Answer: False

This is the definition of regularity (T3), not normality (T4). Normality requires separating any two *disjoint closed sets* — not just a point from a closed set — with disjoint open sets. T4 is strictly stronger. Urysohn's lemma characterizes normal spaces as those where disjoint closed sets can be separated by a continuous real-valued function — a much richer result than what regularity alone provides.

Question 5

Explain why metric spaces are regular. What construction allows you to separate a point from a closed set, and why doesn't this argument generalize to all topological spaces?

Model answer: In a metric space, if x ∉ F and F is closed, then d(x, F) = r > 0 (the distance is positive because F is closed and doesn't contain x). The open ball B(x, r/2) and the open set ∪_{y∈F} B(y, r/2) are disjoint open sets separating x from F. This argument fails in general topological spaces because there is no notion of distance — you cannot form balls of a given radius, and there is no guarantee the 'gap' between a point and a closed set can be turned into an open neighborhood.

This is why regularity appears as a hypothesis in metrization theorems: it captures in purely topological language the key metric-space property of having positive distance from any point to any closed set not containing it. The axiom isolates what metric spaces can do, so that spaces satisfying the axiom are candidates for having a metric assigned to them.