Winter 2016

**Instructor:** Brendon Rhoades

**Instructor's Email:** bprhoades (at) math.ucsd.edu

**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

** Syllabus:**
Please read carefully.

** 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.