Theory Inleiding topologie (WISB 243)

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Inleiding topologie, period 2

1a. Week 46 (14 nov):
  • 1.1: basic ideas of topology, the idea of a topological space,
  • 1.2 ~ 6: specific examples; circle, sphere, Moebius band, torus
  • 1b. Week 46 (17 nov):
  • 1.7, 1.8: Klein bottle, the real projective plane.
  • 1.9: constructions of spaces by quotients and gluing
  • 1.10: metric aspects
  • 2a. Week 47 (21 nov):
  • chapter 2: sect 2.1: definition of topological space, extreme examples
  • example of metric space, metrizability, Euclidean space
  • restriction topology , relative open and closed,
  • sect 2.2: continuous map, homeomorphism, embedding.
  • 2b. Week 47 (24 nov):
  • 2.3: neighbourhoods, basis, basis of neighborhoods, convergent sequences
  • sequential continuity and first countability
  • 2.4: closure, interior and boundary
  • limit point and first countability
  • 3a. Week 48 (28 nov):
  • 2.5: Hausdorffness, metric space is Hausdorff, second countability
  • topological manifold, embedding
  • 2.6: separated subspaces, separation by functions,
  • normal space.
  • 3b. Week 48 (1 dec):
  • 3.1 Quotient topology
  • 3.2 Examples of quotients: abstract torus, Moebius band, Klein Bottle
  • 3.3 Quotients by group actions.
  • 4a. Week 49 (4 dec):
  • 3.4 The n-dimensional real projective space
  • 3.5 product topologies
  • 3.7 basis for a topology, topology basis and second countability
  • metric space with countable dense subspace is second countable
  • 4b. Week 49 (8 dec):
  • non-metrizability of the left limit topology on R
  • 3.6 quotients by collapsing a subspace, examples
  • 3.7 generating topologies
  • skipped: 3.8 topologies on function spaces.
  • 5a. Week 50 (11 dec):
  • 4.1.1: connectedness, definition, definition path connected, examples, connectedness of the unit interval.
  • 4.1.2: basic properties: invariance of connectedness; path connected implies connected
  • 4.1.3: connected components
  • 4.1.4: local pathwise connectedness and connectedness imply pathwise connectedness
  • 5b. Week 50 (15 dec):
  • 4.2.1: definition of compactness, by coverings; the closed unit interval
  • 4.2.2: topological properties of compactness, closed subspace
  • 4.2.3: compactness of products, compactness in R^n
  • Thm. 4.25: invariance under continuous functions
  • 6a. Week 51 (18 dec):
  • Prop. 4.19: separation of disjoint compact subsets;
  • Cor 4.20: normality of compact spaces.
  • 2.4: embedding of a compact space and application to quotients
  • Definition 2.50: topological manifold
  • 2.5: embedding of a compact topological manifold into Euclidean space.
  • 2.6: sequential compactness, equivalence for metrizable spaces.
  • 6b. Week 51 (22 dec):
  • 3.1: local compactness and exhaustion by compact sets.
  • 3.2: the one point compactification.


  • Week 52: this week no lecture (Christmas holiday)
    Week 1 : this week no lecture (Christmas holiday)



    7a. Week 2 (9 jan):
  • 8.1: the space C(X) with its uniform topology and Banach algebra structure.
  • 8.2: the Stone-Weierstrass theorem, formulation and first part of the proof.
  • ** see also the new version of the Extra Notes in blackboard.
  • Mention of the Urysohn Lemma Thm. 5.21, at this point without proof.
  • 7b. Week 2 (12 jan):
  • 8.2: completion of the proof of Stone-Weierstrass
  • 8.3: the formulation of Gelfand-Naimark, and first part of the proof.
  • 8a. Week 3 (16 jan):
  • 8.3: completion of proof of the Gelfand-Naimark theorem.
  • 8.3: algebras, ideals, quotients, spectra.
  • 8b. Week 3 (20 jan): last lecture Inleiding topologie
  • 5.5: Urysohn's lemma and proof;
  • 7.1: Urysohn metrization theorem and proof.

  • 9a. Week 4 (23 jan): no lecture, but question and rehearsal session

    9b. Week 4 (27 jan): no lecture, but question and rehearsal session

    10 Week 5: exam week

    Last change: 10/12-2016