Metric spaces and point-set topology

Builds on

Required prerequisites, all the way down: 36

You will learn to

• Prove that ε–δ continuity between metric spaces agrees with the open-set definition.
• Prove the Heine–Borel theorem, and that continuous images of compact (respectively connected) spaces are compact (connected).
• Construct quotient spaces such as the circle, torus and Möbius band from intervals and squares.

Definition

• A metric on X is a function d : X × X → [0, ∞) with d(x, y) = 0 exactly when x = y, d(x, y) = d(y, x), and the triangle inequality.
• A topology on X is a collection of subsets (the open sets) containing ∅ and X and closed under arbitrary unions and finite intersections.
• A space is compact if every open cover has a finite subcover.
• A homeomorphism is a continuous bijection whose inverse is continuous.

Theorem

• Heine–Borel: a subset of ℝⁿ is compact if and only if it is closed and bounded.
• A continuous real-valued function on a non-empty compact space attains a maximum and a minimum.
• Tychonoff: every product of compact spaces is compact; over ZF this is equivalent to the axiom of choice.
• The interval [0, 1] is connected.

Common misconceptions

• Thinking 'closed' means 'not open'; a set can be both, or neither.
• Assuming a continuous bijection is a homeomorphism; t ↦ (cos t, sin t) from [0, 2π) to the circle is not.
• Believing that closed and bounded implies compact in every metric space.

Leads to

Optional discoveries

• Universal properties Product and quotient topologies are characterised by universal properties: they are the coarsest, respectively finest, topologies making the relevant maps continuous.
• ← Relations, equivalence relations and partitions

Learning pathways

• Undergraduate mathematics major
• Graduate preparation

Further reading

• James R. Munkres, Topology