1 Metric Spaces

26 January 2026 · Denis Auroux

About the course

The first half of the course is topology plus real analysis: (1) point-set topology, i.e. topological spaces (including some pieces of analysis); (2) an introduction to algebraic topology: fundamental groups; (3) a bit more real analysis. Then we move on to complex analysis. Books you should have: Munkres, Topology, 2nd ed.; Ahlfors, Complex Analysis, 3rd ed.; recommended: Rudin, Principles of Mathematical Analysis.

What is topology?

Unlike geometry, which concerns quantitative information about spaces (distances, volumes, …), topology concerns itself with qualitative properties that are invariant under continuous deformation. E.g.: is it connected (a single piece)? Simply connected? (A sphere vs. a torus.)

Point-set topology also gives a language (topological spaces, open and closed sets, compactness, …) both for algebraic topology (which associates algebraic invariants to spaces, e.g. the fundamental group) and for analysis.

Example 1.1. The extreme value theorem says: if is continuous, then achieves its max and min at some points of .

This is in fact true for any continuous whenever is a compact topological space, and is a special instance of the following theorem.

Theorem 1.2. If is a continuous mapping between topological spaces, and is compact, then is compact.

Metric spaces

Since the general notion of topological space is quite abstract, let’s start with a more familiar class of examples: metric spaces (cf. handout; also Rudin, beginning of Chapters 2, 3, 4).

Definition 1.3. A metric space is a set together with a distance function such that:

for , ;
for , ;
for , (triangle inequality).

Example 1.4. with the Euclidean distance .

Example 1.5. If then is a metric space (“induced metric”).

Example 1.6. Different metrics on :

Exercise 1.7. Check that and are metric spaces. What do balls look like?

Definition 1.8. Let be a metric space, , . The open ball of radius around (or neighborhood) is

Here’s a more general notion:

Definition 1.9. is open if , such that .

Proposition 1.10 ((HW!)). Open balls are open; so are arbitrary unions and finite intersections of open sets.

In fact, open sets are unions of open balls! (.)
This is useful for a general discussion of continuity.

Continuity

Definition 1.11. Let , be metric spaces. is continuous if , , such that

Theorem 1.12. is continuous iff open, is open.

Proof.

Assume is continuous, let be open, and let , i.e. . Since is open, such that . By continuity, such that . Hence . So is open.
Conversely, assume open open. Fix , . is open in , so is open in . Hence such that . This means

∎

(Our first – proof, but not our last!)

Sequences and limits

We can also talk about sequences and their limits.

Definition 1.13. A sequence in converges to a limit (write or ) if such that , .

The limit is unique if it exists. Here the get closer to ! Compare with sequences whose terms get closer to each other:

Definition 1.14. A sequence in is Cauchy if such that , .

Exercise 1.15. If a sequence converges then it is Cauchy, but not necessarily vice versa.

A metric space is complete if every Cauchy sequence converges.

Example 1.16. is complete, but (with the induced metric) isn’t complete.

The notion of Cauchy sequence is specific to metric spaces, but really useful for real analysis.

Example 1.17. . If we take this to be the definition of , we can’t prove directly that converges to ; instead, use the Cauchy criterion to show that the limit exists.

Notes not reached in this lecture, covered at the start of Lecture 2

Interlude: what is ?

Answer: it’s an ordered field (i.e. and compatible with the usual rules) with the least upper bound property: every nonempty subset that admits an upper bound ( such that , ) has a least upper bound (i.e. is an upper bound, and every upper bound for is ).

The l.u.b. property is equivalent to completeness of ; any ordered field with this property is isomorphic to . Constructions of from involve adding the missing elements (irrationals) so that the l.u.b. property / completeness holds; the elements of end up being either the sups of certain subsets of (see e.g. Rudin) or the limits of Cauchy sequences in (see HW).

Back to limits of sequences

Proposition 1.18. If , then every open subset contains for all but finitely many .

Proof. , open such that . So such that . ∎

This will be the definition of limit outside the metric case.

Closed sets

Definition 1.19. is closed if its complement is open.

Warning: most subsets of are neither open nor closed!! And… and are both!

Proposition 1.20. If is closed, then for every sequence in which converges to a limit , we have .

Proof. By contradiction. Assume and with . Then open, contains for all but finitely many . However, if is closed then is open, but does not contain for any (since ). This is a contradiction. ∎

(The converse is true in metric spaces and in nice enough topological spaces: “first countable”.)

Topological spaces

Our goal will be to reformulate/generalize all this in the context of topological spaces, i.e. sets equipped with a topology which may or may not come from a metric.

Definition 1.21. A topology on a set is a collection of subsets of , which we’ll declare to be the open sets in . It needs to satisfy the axioms:

, ;
any union of elements of is in ;
the intersection of finitely many elements of is in .

Why bother? One answer: many natural topologies do not come from a metric! E.g., in analysis:

On the space of (bounded) functions , the uniform convergence topology ( iff ) comes from a metric (), but pointwise convergence ( iff , ) doesn’t (“product topology”).
The topology on smooth functions doesn’t come from a metric either.

And on the other hand, a metric contains extraneous information for topology. E.g. , , have the same open sets, hence the same topology.