Math 190: Introduction to Topology
Winter 2016

Instructor's Office: 7250 APM
Instructor's Office Hours: 12:00-1:00pm MF, 2:00-3:00W, or by appointment
Lecture Time: 10:00-10:50am MWF
Lecture Room: 111 SOLIS
TA: Garrett Williams
TA's Email: g2willia (at) ucsd.edu
TA's Office: 2313 APM
TA's Office Hours: Tu, 12:00pm-2:00pm, F 12:00pm-1:00pm
Discussion Time: 8:00-8:50am Tu
Discussion Room: 5402 APM
Final Exam Time: Friday, 3/18/2016, 8:00-10:59am
Final Exam Room: TBA

Lectures:

1/4: Administrivia. What is topology? Examples of topological spaces. Open sets in the standard topology on R^n. Section 12. Definition of a topological space. Topologies on X = {a, b, c}. Proof that the Euclidean topology on R^n is a topology.
1/6: Section 12. The discrete and indiscrete topologies. The finite complement topology. The "finer than" relation on topologies. Section 13. A basis for a topology. The topology generated by a basis. Examples with R^2. The lower limit topology R_l. Proof that the topology generated by a basis is a topology.
1/8: Class cancelled. Prof. Rhoades is in Seattle.
1/11: Section 13. The topology T generated by a basis B consists of all unions of sets in B. Criterion for a collection of open sets to be a basis for a given topology. Criterion for fineness of topologies in terms of bases. The K-topology on R. A subbasis S for a topology on a set X, and the topology which it generates.
1/13: Section 14. Ordered sets. Dictionary order on X x Y. Largest and smallest elements. Intervals. The order topology on an ordered set. Well ordered sets. Axiom: Every set has a well ordering. The minimal uncountable well ordered set S_{Omega}. Every countable subset of S_{Omega} has an upper bound in S_{Omega}.
1/15: Section 14. The dictionary order topology on R x R is strictly finer than the standard topology. Section 15. Definition of the product topology on X x Y. A product of bases is a basis of the product. Examples: R x R, R x R_l. The product topology on a finite product X_1 x ... x X_n of topological spaces. Section 16. The subspace topology on a subset A of a topological space X. An intersection of a basis for X with A is a basis for the subspace topology on A. Examples of subspaces of Euclidean space. Subspaces and products commute.
1/20: Section 17. Closed sets. Closed sets and subspaces. The closure of a set. Criterion for a point to lie in the closure of a set. Examples of closures.
1/22: Section 17. Limit points. Closures and limit points. Sequences and convergence. Hausdorff spaces. A sequence in a Hausdorff space has at most one limit. Finite subsets of Hausdorff spaces are closed. T_1 spaces. Section 18. Definition of a continuous function.
1/25: Section 18. Continuous functions. The Category of Topological Spaces and Continuous Maps. Examples and non-examples. Basic properties of continuous functions. Homeomorphisms.
1/27: Section 18. Imbeddings. More properties of continuous functions. Examples of homeomorphisms. Products and continuity.
1/29: Midterm 1.
2/1: Section 19. Infinite products of sets. The product and box topologies on an infinite product of spaces. The Universal Property of the Product Topology. Anonymous course feedback.
2/3: Sections 20/21. Metrics on sets. The metric topology. Metrizable spaces: examples and non-examples.
2/5: Quotient Topology Supplement. Review of equivalence relations. Examples. The definition of the quotient topology on X/~. The canonical projection from X to X/~ is continuous. Universal property of the quotient topology (statement).
2/8: Proof of the universal property of the quotient topology. Example of homeomorphism involving a quotient space. Some examples of quotient spaces.
2/10: Quotient maps. Examples and non-examples. Section 23. Separations. Connected topological spaces. Continuous images of connected spaces are connected. Connectivity is a topological invariant. A union of connected subspaces is connected if these subspaces have non-empty intersection.
2/12: Section 24. Linear continuums. A linear continuum L (or a convex subset thereof) is connected in the order topology. Path connected spaces. Path connected implies connected, but not conversely. Intermediate Value Theorem and meteorological example. Section 25. Local (path) connectivity.
2/17: Section 25. Components and path components of spaces. Properties of components and path components. Section 26. Covers. Subcovers. Open covers. Compact spaces.
2/19: Section 26. Examples and non-examples of compact spaces. A closed subspace of a compact space is compact. A compact subspace of a Hausdorff space is closed. If f: X to Y is a continuous bijection with X compact and Y Hausdorff, then f is a homeomorphism. A finite product of compact spaces is compact. Statement of the Tychonoff Theorem.
2/22: Section 27. If X is an order topology with the least upper bound property, closed intervals in X are compact. Heine-Borel Theorem. Extreme Value Theorem. Lebesgue Number Lemma.
2/24: Midterm 2.
2/26: Section 27. Uniform continuity of maps between metric spaces. Section 28. Limit point compactness. Sequential compactness.
2/29: Section 28. Compactness, limit point compactness, and sequential compactness are equivalent for metric spaces. Section 29. Local compactness.
3/2: Prof. Rhoades is in Los Angeles. Prof. Novak will discuss Section 29. The one-point compactification of a locally compact Hausdorff space.
3/4: Section 30. Countable bases at points. First-countable spaces. Second-countable spaces. Dense subsets of spaces. Separable spaces. Lindeloef spaces. Examples and non-examples.
3/7: Sections 31 and 32. Regular spaces. Normal spaces. Subspaces and products of regular spaces are regular. Subspaces and products of normal spaces need not be normal. Compact Hausdorff spaces are normal. Metrizable spaces are normal.
3/9: Section 33. The Urysohn Lemma and its proof. Completely regular spaces. Axiomania.
3/10: Bonus lecture! The fundamental group of a space X.
3/11: Section 34. The Urysohn Metrization Theorem and its proof.

Homework Assignments:

Homework 1, due 1/13/2016.
Homework 2, due 1/20/2016.
Homework 3, due 1/27/2016.
Homework 4, due 2/3/2016.
Homework 5, due 2/10/2016.
Homework 6, due 2/17/2016.
Homework 7, due 3/2/2016.
Homework 8, due 3/9/2016.

Optional Homework.

Midterms:

Practice Midterm 1.
Midterm 1.
Score Distribution: 100, 97, 96, 94, 93, 92, 91, 91, 85, 84, 83, 83, 81, 79, 78, 75, 75, 71, 56, 55.

Practice Midterm 2.
Midterm 2.
Score Distribution: 99, 91, 89, 87, 87, 86, 84, 83, 82, 73, 71, 69, 68, 60, 60, 55.

Practice Final.