Fall 2014

**Instructor:** Brendon Rhoades

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

**Instructor's Office:** 7250 APM

**Instructor's Office Hours:** 10:00-11:00am MWF or by appointment

**Lecture Time:** 12:00-12:50pm MWF

**Lecture Room:** B-104 MANDE

**TA:** Jay Cummings

**TA's Email:** jjcummings (at) ucsd.edu

**TA's Office:** 6321 APM

**TA's Office Hours:** 9:00am-10:00am M and 12:00pm-1:00pm Th

**Discussion Time:** 7:00-7:50pm W

**Discussion Room:** B402A APM

**Final Exam Time:** Thursday, 12/18/2014, 11:30am - 2:29pm

**Final Exam Room:** TBA

** Syllabus:**
Please read carefully.

** Lectures:**

**10/3:** Administrivia. What is topology? Open sets in R^n.
Section 12. Definition of a topological space. Topologies on
X = {a, b, c}.

**10/6:** Section 12.
Examples of topologies:
The Euclidean topology on R^n. The
discrete and indiscrete topologies on any set X. The finite complement
topology on any set X.
The definition of one topology being "finer than" another topology on
the same set X.

**10/8:** Section 13. The basis for a topology on a set X;
the topology generated by a given basis. If B is a basis for a topology T
on a set X, then T = {all unions of sets in B}.

**10/10:** Section 13. If a collection **C** of open sets
in a topological space X satisfies the property that for any open set U
in X and any x in U, there is some C in **C** such that x is in C is
contained in U, then **C** is a basis for the topology of X.
Criterion for relative fineness of topologies coming from bases.
The lower limit topology on R.

**10/13:** Section 13. The subbasis for a topology.
Section 14. Ordered sets. Examples of orders: Subsets of R under
ordinary order <, dictionary order. The order topology on an ordered set.

**10/15:** Section 15. The product topology on X x Y for topological
spaces X and Y. Section 16. The subspace topology on a subset A of
a topological space X. Some subspaces of R^n (spheres, hypercubes, the torus).

**10/17:** Prof. Rhoades was in Halifax; Prof. Rogalski covered
closed sets, limit points, closures, Hausdorff spaces, and sequences
and convergence. This material is Section 17.

**10/19:** Prof. Rhoades was in Halifax; see above.

**10/22:** Section 17. A sequence in a Hausdorff space converges to
at most one point. Section 18. Definition of a continuous function.
Statement that continuity of functions between Euclidean spaces is equivalent
to the "epsilon-delta" definition. Definition of a homeomorphism.

**10/24:** What is topology, Part II.
Section 18. Characterizations of continuous functions.
Imbeddings. Examples: f(x) = exp(x) is an imbedding from R to R.
g(x) = tan(pi*x/2) is a homeomorphism from (-1, 1) to R.
The map h(t) = (cos(2*pi*t), sin(2*pi*t) is a conintuous bijection from
[0,1) to S^1, but not a homeomorphism. (Are [0,1) and S^1 homeomorphic?)

**10/27:** Section 18. Basic properties of continuous functions between
topological spaces. The Pasting Lemma. Section 19. The box and product
topologies on (possibly infinite) Cartesian products of topological spaces.
The Universal Property of the Product Topology.

**10/29:** Midterm 1.

**10/31:** Section 19. Proof of the Universal Property of the Product
Topology. Section 20. Metrics on sets. Examples: The standard metric on
R^n, a metric on {all continuous functions [-1, 1] --> R}. The metric
topology on a set X with metric d. Metric topologies are Hausdorff. Statement
of the Ulrysohn Metrization Theorem. Anomymous course feedback.

**11/3:** Section 20. Bounded metric spaces. Criterion for
relative fineness of topologies coming from two metrics on the same set.
The standard bounded metric attached to a metric. The epsilon-delta
definition of continuity for maps between metric spaces.
The Sequence Lemma. First half of the proof that R^J (J uncountable)
is not metrizable in the product topology.

**11/5:** Section 21. Completion of the proof that R^J (J uncountable)
is not metrizable in the product topology. The sequential characterization of
continuity for maps f: X --> Y with X metrizable. Section 22. Equivalence
relations and equivalence classes. The set X/~ for an equivalence relation
~ on a set X. Definition of the quotient topology on X/~ for a topological space
X.

**11/7:** Section 22. The Universal Property of the Quotient Topology.
Proving that R/~ (x~y iff x-y in Z) is homeomorphic to S^1 (using the Universal Property).
The definition of a quotient map p: X --> Y.

**11/10:** Section 22.
Quotient maps, saturated sets, open maps, and closed maps. Examples.
The quotient space X/A for a subspace A of X.
Examples. R/Q is not Hausdorff. Section 23.
Separations and connected spaces. Clopen sets. Examples and non-examples.

**11/12:** Section 23. Continuous images of connected spaces are connected.
Unions of connected subspaces {C_a} such that the C_a have nonempty intersection
are connected. If X and Y are connected, so is X x Y.
Section 24. Supremums, the completeness axiom for R.
R, intervals in R, and rays in R are all connected.

**11/14:** Section 24. Linear continua. Paths. Path connectedness.
Every path connected space X is connected. Topologist's comb and sine curve.
Section 25. Components and path components of a space X.

**11/17:** Section 25. Local connectedness and local path connectedness.
In a locally path connected space X, the components and path components
coincide. Section 26. Covers and open covers. Compactness.
R and [0,1) are not compact. Any finite space is compact.
Closed subspaces of compact spaces are compact.

**11/19:** Section 26. Compact subspaces of Hausdorff spaces
are closed. Continuous images of compact spaces are compact.
If X and Y are compact, so is X x Y.
Statement of the Tychonoff Theorem.

**11/21:** Section 27. [0,1] is compact. Heine-Borel Theorem.
Extreme Value Theorem. Lebesgue Number Lemma. Uniform Continuity.
If X and Y are metric spaces and X is compact, a continuous function
X --> Y is uniformly continuous.
**11/24:** Midterm 2.

**11/26:** Gobble gobble.

**12/1:** Section 28. Limit point compactness.
(0,1] is not limit point compact as a subspace of R.
[0,10] intersect Q is not limit point compact as a subspace of R.
Every compact space is limit point compact.
Subsequences and sequential compactness.
If X is metrizable, TFAE:
(1) X is compact, (2) X is limit point compact, (3) X is sequentially compact.
The Bolzano-Weierstrass Theorem.

**12/3:** Section 29. Local compactness. R is locally compact.
Q is not locally compact. One-point compactifications.

**12/5:** Section 30. First Countability, Second Countability,
Separability, and the Lindeloef property.
Examples: R^n is second countable (and so first countable,
separable, and Lindelof). An uncountable set X in the discrete topology
is not separable or Lindeloef (so not second countable), but is first countable.
An uncountable set X in the finite complement topology is
not first countable (so not second countable), but is compact (hence Lindeloef)
and separable.

**12/8:** Sections 31 and 32. Regular and normal spaces. The K-topology
R_K on R is Hausdorff but not regular. Statement that the
product space R_l x R_l is regular but not normal. Every metric space is normal.
(By homework, every compact Hausdorff space is normal.)
Statement of Ulrysohn's Lemma.

**12/10:** Section 33. Reformulation of normality: A space X is normal
if and only if for every closed subset A of X and any neighborhood U of A,
there exists a neighborhood V of A such the the closure of V is contained in U.
The proof of Ulrysohn's Lemma. Completely regular spaces.

** Homework Assignments: **

Homework 1, due 10/10/2014.

Homework 2, due 10/17/2014.

Homework 3, due 10/24/2014.

Homework 4, due 11/7/2014.

Homework 5, due 11/14/2014.

Homework 6, due 11/21/2014.

Homework 7, due 12/5/2014.

Homework 8, due 12/12/2014.

** Midterms: **

** Midterm 1 ** Score distribution (out of 115):
110, 110, 109, 105, 97, 90, 89, 89, 82,
81, 68, 68, 67, 58, 45, 36, 32.

Exam and
Solutions.

** Midterm 2 ** Score distribution (out of 95):
94, 91, 82, 80, 79, 75, 58, 54, 51, 47, 42,
42, 38, 36, 29, 27, 26.

Exam and
Solutions.