Inleiding topologie, period 2

1a. Week 46 (12 nov):

1.1: basic ideas of topology, the idea of a topological space,

1.2 ~ 7: specific examples; circle, sphere, sterographic projection, Moebius band, torus, Klein bottle

1b. Week 46 (15 nov):

1. 8: The projective plane

1.9: quotient by an equivalence relation, abstract and associated to a glueing map (see the extra lecture notes in blackboard).

1.10: open sets in a metric space, criterion for continuity. Compactness and completeness are deferred to a later lecture.

Definition 2.1 of topology.

2a. Week 47 (19 nov):

2.1: Examples of topologies, metrizable topology, subspace topology

2.2: continuity of a map, calculus of preimages, homeomorphism, embedding.

2.3: neighborhood, collection of neighborhoods of a point, continuity at a point

2b. Week 47 (22 nov):

2.3: sequential continuity versus continuity, basis of neighborhoods of a point, first countability axiom.Thm 2.29 and 2.34.

2.4: the notions of open, closed, boundary

2.5: Hausdorff property, uniqueness of limits of sequences

3a. Week 48 (26 nov):

2.5: second countability, topological manifolds.

postponed: 2.6: more on separation, normality, metric spaces are normal, Urysohn metrizability thm mentioned

3.1: the quotient topology, continuity of the induced map.

3.2: examples of quotients; when induced map is embedding

3b. Week 48 (29 nov):

3.3 Quotients by group actions, Hausdorffness preserved,

3.4 The projective space realized as a topological quotient

4a. Week 49 (3 dec):

3.5 Product topologies

3.6 special classes of quotients II: collapsing, cone, suspension, simplex

3.7 topology basis, Prop. 3.17

3.8 topology generated by a collection of subsets, basis of generated topology

4b. Week 49 (6 dec):

3.8 initial topologies for a collection of maps

3.9 examples of initial topologies, topologies on spaces of functions.

4.1.1 the notion of connectedness, proof that [0,1] is connected.

4.1.2: basic properties of connectedness

5a. Week 50 (10 dec):

4.1.2 basic properties of connectedness, continued

4.1.3 connected components,

4.1.4 connectedness versus pathwise connectedness

4.2.1 compactness, [0,1] is compact.

5b. Week 50 (13 dec):

4.2.2 topological properties of compact spaces, normality

2.6 normality of metric spaces

4.2.3 compactness of products,

4.2.4 compactness under continuous functions.

6a. Week 51 (17 dec): the lecture will be given by Marius Crainic.

4.2.5: embeddings of compact topological manifolds

4.2.6: sequential compactness

4.3.1: locally compact Hausdorff spaces, Exercise 4.10, 2nd countable case: existence of exhaustion by compacts

6b. Week 51 (20 dec): the lecture will be given by Marius Crainic.

4.3.2: the one point compactification of a locally compact Hausdorff space

4.3.2: on second countability

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Week 52: this week no lecture (Christmas holiday)

Week 1 : this week no lecture (Christmas holiday)

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7a. Week 2 (7 jan):

5.1 and 5.2: finite partitions of unity, Lemma 5.3 and Thm 5.6.

5.3. locally finiteness, refinement, paracompactness (definitions).

7b. Week 2 (10 jan):

5.3, paracompactness and normality (proof of Lemma 5.17).

Lemma 5.17: shrinking lemma.

arbitrary partitions of unity: Theorem 5.18.

5.4, the locally compact case, Thm 5.20

Thm 5.19, criterion for normality of a family of functions.

8a. Week 3 (14 jan):

Completeion of Thm 5.19

5. Urysohn's lemma, Thm 5.21.

8b. Week 3 (17 jan): last lecture Inleiding topologie

7.1: Urysohn metrization theorem;

7.2: Smirnov metrization thoerem.

for the proofs of both, see the extra notes in blackboard, course content.

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9a. Week 4 (21 jan): no lecture, but question and rehearsal session

9b. Week 4 (24 jan): no lecture, just exercise class.

10 Week 5: exam week

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Last change: 22/11 - 2018