Inleiding topologie, period 2

1a. Week 46 (13 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 (16 nov):

1.7, 1.8: Klein bottle, the real projective plane.

1.9: constructions of spaces by quotients and gluing; see the extra notes

1.10: metric aspects

2a. Week 47 (20 nov):

2.1: topological space, open and closed, examples

2.2: definition of continuous map, homeomorphism, topological property

2b. Week 47 (23 nov):

2.3: neighborhood of a point, continuity at a point

convergent sequence, sequential continuity
basis of neighborhoods, first countability.

2.4: interior, closure and boundary.

3a. Week 48 (27 nov): het college wordt gegeven door Dr. Gijs Heuts

2.5: notion of Hausdorffness; metric space is Hausdorff.

basis of a topological space and second countability.

definition of a topological manifold, with example

2.6: topological separation vs separation by functions

the notion of normality.

3b. Week 48 (30 nov): het college wordt gegeven door Dr. Gijs Heuts

3.1: quotient topology, definition and continuous map with quotient as domain continuous maps on the quotient. Hausdorffness may be lost in quotient.

3.2: continuity of the map that produces a realization of the abstract quotient.

3.3: quotient for continuous action of finite group. If time permits: Hausdorffness is preserved.

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 (7 dec):

3.6 quotients by collapsing a subspace, examples

3.7 generating topologies

3.8 topologies on function spaces.

5a. Week 50 (11 dec): cancelled because of snow conditions

5b. Week 50 (14 dec):

read by yourself: 3.8: the topology of pointwise convergence, and:

the topology of uniform convergence is metric (at a suitable moment, we will return to these subjects).

4.1.1: connected and path connected, definition and examples

unit interval is connected

4.1.2: connectedness and image under continuous map

path connected implies connected, the removing a point trick

4.1.4: connected versus path connected

4.1.3: connected components, beginning

6a. Week 51 (18 dec): the second hour lecture will be given by Dr. Gijs Heuts.

4.1.3: connected components, completion.

4.2.1: open cover, defi of compactness

4.2.2: proof of Prop 4.18, Thm 4.19, Prop. 4.20 and normality of compact Hausdorff space

4.2.3 compactness of products, compactness in R^n

4.2.4 compactness preserved under continuous maps

6b. Week 51 (21 dec):

4.2.4: Corollary 4.28 and application to concrete realization of quotients.

4.2.5 embeddings of compact manifolds

4.2.6 sequential compactness

4.3.1 local compactness; the existence of an exhaustion

4.3.2 the one point compactification

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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 (8 jan):

4.3.2: continuation of one point compactification

5.2: finite partitions of unity

5.1: axioms for sets of functions

7b. Week 2 (11 jan):

finishing finite partitions of unity

5.3 arbitrary partitions of unity: paracompactness

5.4 mention of Theorems 5.19, 5.20.

8a. Week 3 (15 jan):

completion Chapter 5: 5.3.

5.5 Urysohn's lemma

Thm 6.11: every metric space is paracompact, just mentioned.

8b. Week 3 (18 jan): last lecture Inleiding topologie

7.1: The Urysohn metrization theorem.

7.2: The Smirnov metrization theorem.

7.3: Consequences of 7.1, 7.3.

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9a. Week 4 (22 jan): no lecture, but question and rehearsal session

9b. Week 4 (25 jan): no lecture, just exercise class.

10 Week 5: exam week

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Last change: 23/1 - 2018