Schedule

This is the tentative course schedule; there may be small changes as the semester progresses.


Week 1 (starts Jan 10)

First day of class: Wednesday, Jan 12.

Sections and topics covered:
§ 1.3, 1.4: Review of basic set theory and set operations.

Week 2 (starts Jan 17)

No class on Monday, Jan 17 (MLK Day).

Sections and topics covered:
§ 1.5: Cartesian products of sets.
§ 1.6: Basic properties of functions.

Week 3 (starts Jan 24)

Sections and topics covered:
§ 2.1: Intervals, upper and lower bounds in \(\mathbb{R}\).
§ 2.2: Finite and infinite sets, cardinality.

Week 4 (starts Jan 31)

Sections and topics covered:
§ 2.1: The density of the rational numbers as a subset of \(\mathbb{R}\).
§ 2.3: The topology of \(\mathbb{R}\): open sets in \(\mathbb{R}\).

Week 5 (starts Feb 7)

Sections and topics covered:
§ 3.1: Definition and examples of metric spaces, including the plane \(\mathbb{R}^2\).
§ 3.2: The topology of metric spaces: open and closed sets in metric spaces.

Week 6 (starts Feb 14)

Sections and topics covered:
§ 3.2: The topology of metric spaces: open and closed sets in metric spaces.
§ 3.3: Interior, closure and boundary in metric spaces.
§ 3.4: Continuous functions between metric spaces.

§ 3.7: Complete metric spaces


Week 7 (starts Feb 21)

Sections and topics covered:
§ 3.4: Continuous functions between metric spaces.
§ 3.5: Equivalence of metric spaces
§ 3.6: New spaces from old

Week 8 (starts Feb 28)

In-class midterm: Monday, Feb 28.

Sections and topics covered:
§ 3.6: New spaces from old
§ 4.1: Topological spaces — definition and examples

Week 9 (starts Mar 14)

Sections and topics covered:
§ 4.1: Topological spaces — definition and examples
§ 4.2: Interior, closure, and boundary
§ 4.3: Basis and subbasis

Week 10 (starts Mar 21)

Sections and topics covered:
§ 4.3: Basis and subbasis
§ 4.4: Continuity and topological equivalence

Week 11 (starts Mar 28)

Sections and topics covered:
§ 4.5: Subspaces
§ 5.1: Connected and disconnected spaces
§ 5.2: Theorems on connectedness

Week 12 (starts Apr 4)

Sections and topics covered:
§ 5.2: Theorems on connectedness
§ 5.5: Path connected spaces

Week 13 (starts Apr 11)

Sections and topics covered:
§ 5.5: Path connected spaces
§ 5.6: Locally connected and locally path-connected spaces
§ 6.1: Compact spaces and subspaces

Week 14 (starts Apr 18)

Sections and topics covered:
§ 6.2: Compactness and continuity
§ 6.3: Properties related to compactness (Selected topics — see lecture notes)
§ 6.5: The Cantor set