Metric Spaces
Sets with a distance function.
Key idea
Sets with a distance function.
In depth
A metric d(x, y) generalises distance with positivity, symmetry and the triangle inequality. It lets you talk about convergence, open sets and continuity in settings such as function spaces.
Worked example
On R², d₁ (taxicab), d₂ (Euclidean) and d∞ (max) are all metrics, with different "balls".
Formula
Real-world uses
- Measuring distance between DNA sequences, images or texts in data science
Common mistake
Assuming a metric must be Euclidean.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Metric Spaces?
Sets with a distance function. It is a topic at the expert level in Analysis & Topology.
What is the formula for Metric Spaces?
The key formula is d(x,z) ≤ d(x,y)+d(y,z).
Where is Metric Spaces used in real life?
Real-world examples: Measuring distance between DNA sequences, images or texts in data science.
What should I learn before Metric Spaces?
Make sure you are comfortable with Fourier Series Convergence, the previous topic in Analysis & Topology.
What is a common mistake with Metric Spaces?
Assuming a metric must be Euclidean.
How can I practise Metric Spaces?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Metric Spaces?
Move on to Open and Closed Sets next.
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.