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Metric spaces and point-set topology

From metric spaces to topological spaces: open and closed sets, continuity, homeomorphism, connectedness, compactness, product and quotient topologies, and separation axioms.

math.geom.point-set-topology

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Se basa en

Real analysis

Requisitos previos obligatorios, hasta el fondo: 36

Aprenderás a

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.

Definición

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.

Teorema

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.

Errores frecuentes

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.

Conduce a

Homotopy and the fundamental group Knot theory Descriptive set theory Smooth manifolds Algebraic geometry Sheaves Measure theory and integration

Descubrimientos opcionales

Universal properties Product and quotient topologies are characterised by universal properties: they are the coarsest, respectively finest, topologies making the relevant maps continuous.
Surfaces, polyhedra and the Euler characteristic Compact surfaces are the first rich family of topological spaces to classify completely.
← Relations, equivalence relations and partitions

Itinerarios de aprendizaje

Lecturas complementarias

James R. Munkres, Topology