Topology · Chapter 01

Metric spaces and convergence

What a distance has to do, the balls and open sets it produces, and what convergence means once the only tool is a distance.

d (x, z) \leq d (x, y) + d (y, z)

B_{r} (x)

d (x, y) = d (y, x)

d (x, x) = 0

x_{n} \to x

d (x_{n}, x) \to 0

d ∣_{Y \times Y}

ε > 0

Lessons

Build the foundation

01DistanceTopology begins by asking what a distance really has to do. Very little, it turns out: three conditions are enough, and a great many different rules satisfy them.Read lesson →
02Examples of metricsOne set carries many metrics. On the plane the straight-line distance is only the most familiar: a taxi driver measures along the streets, and a walker who only minds the worst leg of the journey counts something else again. Each rule satisfies the three conditions, so each is a metric, and each decides what near is going to mean.Read lesson →
03Open and closed ballsOnce there is a distance, the natural thing to collect is every point closer to a chosen point than some fixed amount. That collection is called a ball, and it is the basic shape of the whole subject.Read lesson →
04Open sets in a metric spaceA set is called open when none of its points is on the edge: around every one of them there is a ball, however small, that the set contains entirely.Read lesson →
05Unions and intersectionsOpen sets combine freely in one direction and only sparingly in the other. Any number of them may be united and the result is open. Only finitely many may be intersected.Read lesson →
06

Convergent sequences

A sequence converges to a point when its terms end up inside every ball around that point, however small the ball is asked to be. Only the distance is used, so the definition reads the same in the plane, in a space of functions and in a set with the discrete metric — and the answers differ wildly.

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07Cauchy sequencesConvergence names a limit. There is a weaker demand that names nothing: that the terms eventually get close to each other. A sequence whose terms huddle together like this is called Cauchy, and it is what you can check when the limit is exactly the thing you are trying to find.Read lesson →

08Complete metric spacesA space is complete when every sequence that tries to converge succeeds — that is, when every Cauchy sequence has a limit inside the space. Completeness is what separates the real line from the rationals, and it is the property everything that approximates depends on.Read lesson →

09Subspaces and induced metricsAny subset of a metric space is a metric space in its own right: keep the same distance rule and simply stop asking about points outside. What changes is not the distances but the balls — and with them which sets count as open, because a ball of the subset is only the part of the old ball that survived.Read lesson →

10Chapter practiceBring the chapter together: check the conditions on a distance, decide what a ball holds, test a set for openness, combine open sets, and separate a neighbourhood from an open set. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Metric spaces and convergence

Build the foundation