Metric Spaces: Distance as an Axiom
Keep only the properties distance must have, and check how much of the analysis of the real line survives on that alone.
What a learner can do afterwards
- State the metric axioms and verify them for an unfamiliar example such as a function space
- Redefine convergence and continuity using only the metric
- Give two metrics on one set that disagree about which sequences converge
1 · Read
A metric turns distance into four axioms: nonnegativity, identity of indiscernibles, symmetry, and the triangle inequality. Anything satisfying all four earns the name, however exotic. Test candidates ruthlessly, since many plausible distances fail symmetry or the triangle part.
The discrete metric puts distinct points one unit apart and identical points at zero. Under the usual metric, 1 over n tends to 0, but discretely it never settles: discrete convergence demands eventual constancy, which a strictly moving sequence lacks.
Convergence and continuity need only a metric. A sequence arrives at x when distances shrink to zero, and a function is continuous at a when nearby points map to nearby values. Sequential continuity matches ball continuity here. On continuous functions the uniform metric takes the maximum gap, and its triangle inequality holds because pointwise control survives taking the maximum.
One set can carry rival metrics that disagree. With radius 0.01 about zero, the first 100 terms of 1 over n sit outside, since 1 over n reaches 0.01 exactly at n equal 100. Arrival belongs to the ruler, not just the points, which opens the door to topology.
Four axioms buy convergence and continuity, and the choice of ruler decides which sequences arrive.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.