Separation Axioms

Key idea

Hausdorff, regular and normal spaces.

In depth

T₁, T₂ (Hausdorff), T₃, T₄ describe how well open sets separate points and closed sets. In a Hausdorff space distinct points have disjoint neighbourhoods, which guarantees unique limits.

Worked example

Metric spaces are Hausdorff, while the Zariski topology generally is not.

Real-world uses

• Ensuring points in a space can be told apart, important in data and network models

Common mistake

Assuming limits are unique in every topological space.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.