Winter 2018

**Instructor:** Brendon Rhoades

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

**Instructor's Office:** 7250 APM

**Instructor's Office Hours:** 10:00-10:50 am MWF

**Lecture Time:** 11:00-11:50pm MWF

**Lecture Room:** 222 CENTR

**TA:** Dun Qiu

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

**TA's Office:** E106 SDSC

**TA's Office Hours:** 9:00-11:00 am Th

**Final Exam Time:** Monday, 3/19/2018, 11:30am-2:29pm

**Final Exam Room:** TBA

** Syllabus:**
Please read carefully.

** Homework Assignments: **

Homework 1, due 1/18/2018.

Homework 2, due 1/25/2018.

Homework 3, due 2/1/2018.

Homework 4, due 2/15/2018.

Homework 5, due 2/22/2018.

Homework 6, due 3/1/2018.

Homework 7, due 3/15/2018.

** Lectures: **

**1/8:** Administrivia. Examples of topological spaces. Section 12. Topologies on sets.
The Euclidean topology on R^n (distances and open balls).
Discrete and indiscrete topologies.

**1/10:** The fineness/coarseness relation on topologies.
Finite complement topology. Section 13. Bases for topologies. The topology generated by a basis;
proof that this is actually a topology. Examples. The lower limit topology
R_{ell} on the real line.

**1/12:** Comparing fineness of topologies generated by two different bases on the
same set. Section 14. Ordered sets. The order topology on an ordered set. The ordered square I^2
(under dictionary order).

**1/17:** Well ordered sets. The minimal uncountable well ordered set. Section 15. The product
topology on a product X x Y of two topological spaces. (And finite products X_1 x ... x X_n
more generally). Bases of spaces give rise to bases of products. Section 16. Subspace
topology. Subspaces and products commute.

**1/19:** Section 17. Closed sets. Closures of sets. Limit points. Neighborhoods.
Sequences and limits in arbitrary spaces. The Hausdorff axiom.

**1/22:** Section 17. In a Hausdorff space limits are unique when they exist.
T_1 spaces. Every Hausdorff space is T_1. Section 18. Continuous functions f: X --> Y between
topological spaces X, Y. Examples. Homeomorphisms.

**1/24:** The circle and the square are homeomorphic. Constant functions are continuous.
Inclusions of subspaces are continuous. [0,1) is homeomorphic to the non-negative
reals. Continuity is preserved under domain restriction or codomain restriction/extension.
Matrix space Mat_n(R) as Euclidean space R^{n^2}. The determinant is a continuous function
on Mat_n(R); GL_n(R) is open in Mat_n(R).

**1/26:** Pasting Lemma. Examples. Projections
out of products. Continuous functions into products. Universal property of X x Y.
Section 19. Arbitrary products of sets. The product and box topologies on an arbitrary
product of spaces.

**1/29:** Projections out of products. The Universal Property of the Product Topology.
Hausdorfness and closure and products. Pathologies of the box topology: continuity
and convergence. Section 20. Metrics on sets. Example.

**1/31:** The metric topology on a set X with a metric d. Examples. Metrizable
spaces. Every metrizable space is Hausdorff. Closure in metrizable spaces
can be detected with sequences; the box topology on R x R x R x ... is not
metrizable.

**2/2:** Every order topology is Hausdorff. The order topology on
S_{Omega} U {Omega} is not metrizable. Extension of sequential
and delta/epsilon notions of continuity to metric spaces.
Extension of uniform limit theorem to metric spaces. Introduction
to quotients.

**2/5:** Midterm 1.

**2/7:** NO CLASS - Prof. Rhoades is in LA.

**2/9:**
Quotient Topology Supplement.
Equivalence relations ~ on a set X. Equivalence classes [x].
The set X/~ of equivalence classes. Canonical surjection onto X/~.
Examples. X/A where A is a subset of X. The definition of the quotient topology
on X/~. Examples when X = R.

**2/12:**
Proof the the quotient topology is a topology.
Maps out of sets X/~ of equivalence relations. The Universal Property
of the Quotient Topology. Proof that R/~ where x ~ y iff x - y is an integer
is homeomorphic to S^1.

**2/14:** Quotient maps. Section 23. Separations. Disconnected
and connected spaces.
Continuous images of connected spaces are connected.
A union of connected spaces which share at least one point
in common is connected.

**2/16:** Connectedness is a homeomorphism invariant.
A product of two connected spaces is connected. Section 24.
Linear continuums. A linear continuum (or any interval or ray therein)
is connected. Application: R^n, S^n, I^n, etc. are connected.
Intermediate Value Theorem.
There exist two antipodal points on the earth with the same temperature.

**2/21:** Section 25. [0,1) and S^1 are not homeomorphic.
Paths in spaces. Path connectivity. Path connected implies connected.
Topologists' comb and sine curve. Components and path components.
Local connectedness and local path connectedness. Examples.
Section 26. Covers, subcovers, open covers. Definition of compactness.
R is not compact. Q (intersect) [0,1] is not compact.

**2/23:** Section 26. Continuous images of compact spaces are compact.
If X and Y are compact, so is X x Y. Section 27. If X is an order
topology with the least upper bound property, any closed interval
[a,b] in X is compact. Heine-Borel Theorem.

**2/26:** Section 27. Extreme Value Theorem. Section 28.
Limit point compactness. Compact implies limit point compact, but
not conversely. Statement that compactness, limit point compactness,
and sequential compactness are equivalent in metrizable spaces X.
Section 29. Local compactness. Statement: Any locally compact
Hausdorff space admits a one-point compactification Y.
Construction of Y. Start of proof of validity.

**2/28:** Wrap-up of one-point compactifications.
Crash course in group theory. Group axioms. Examples:
groups of numbers, GL_n. Group homomorphisms.
Group isomorphisms.

**3/2:** Free groups. Idea of the fundamental groups.
Section 51. Homotopy of maps X to Y. Path homotopy of paths
I to X with the same starting and ending points. Convex subsets
of Euclidean space. Examples.

**3/5:** Midterm 2.

**3/7:** More on path homotopy. Path homotopy classes.
[a] * [b] for path homotopy classes of a, b. Associativity,
identity, and inverse via path homotopy classes.
Loops. Definition of the fundamental group (X, x_0)
of space X based at x_0.

**3/9:** If f: (X, x_0) to (Y, y_0) is a continuous
map between spaces, we get an induced homomorphism
f_*: pi_1(X,x_0) to pi_1(Y,y_0) between fundamental
groups. The fundamental group is a homeomorphism invariant.
A path from x_0 to x_1 induces an isomorphism between
pi_1(X,x_0) and pi_1(X,x_1). Some examples of fundamental
groups. Covering maps p: E to B. Examples.

**3/12:** More on covering maps.
Compact metric spaces and statement of the Lebesgue
Number Lemma.
Lifts.
The path lifting property for covering
spaces. The homotopy lifting property for covering spaces.
Proof that the fundamental group of the circle is isomorphic to the
additive group of integers.

**3/14:** The fundamental group of the n-torus.
The fundamental group of the n-dimensional sphere.
Fundamental groups of projective spaces and bouquets of circles.
Retractions. Application: There is no retraction of the closed disc
onto its boundary circle.

**3/15:** Bonus lecture! Simplicial complexes. Topological realizations.
f-vectors. Krustal-Katona Theorem and shifted complexes.
Simplicial spheres. Euler's formula. Euler characteristic of a complex.
Dehn-Sommerville Equations. The moment curve and cyclic polytopes.

** Lecture Notes: **

Lectures 1-2

Lecture 3

Lectures 4-6

Lectures 7-10

Lectures 11-15

Lectures 16-20

Lectures 21-25

** Midterms: **

Midterm 1

Score Distribution: 98, 97, 94, 90, 88, 81, 79, 77, 76,
76, 74, 72, 71, 70, 67, 61, 57, 55, 55, 53, 49, 46, 46,
45, 45, 44, 37, 27, 26, 21.

Mean: 62.6, Median: 64, Standard Deviation: 21.0

Midterm 2

Score Distribution: 91, 89, 88, 79, 77, 76, 74,
73, 72, 72, 70, 70, 70, 69, 68, 62, 60, 60,
53, 52, 52, 52, 50, 46, 44, 39, 34, 34.

Mean: 63.4, Median: 69.5, Standard Deviation: 15.9