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