MATH 467--HONORS TOPOLOGY--FALL 2019

Syllabus

References

Possible topics for presentation

COURSE LOG:

8/21 W  Prerequisites, def. of metric space, examples

8/23 F  Normed linear spaces (unit balls convex and symmetric) , open sets, limits, equivalent metrics

Problem set 1 (due 8/30)

8/26 M path metrics, topological spaces (def.), discrete metric sp.

8/28 W remarks on path metrics, closed sets (equivalent definitions)

8/30 F continuity (equivalent conditions); composition and homeomorphism (def.)
distance function to a set

Problem set 2 (due 9/6) 1.11 (limit point defined in 1.10), 1.14, 6.2, 6.3 (a)(b)

9/2   M Labor Day (no lecture)

9/4   W examples of homeomorphisms/metrics on function spaces and uniform convergence
Problem set 1-solutions
(Includes also notes for the 9/4 lecture.)

Question to think about: find necessary and sufficient conditions on a metric d on a linear space E,
for d to be equivalent to the metric induced by some norm on E.
(For example, a necessary condition is that translations and dilations of E be homeomorphisms.)

9/6   F Cauchy sequences, completeness: examples of incomplete spaces, completeness of map spaces.

Problem set 3 (due 9/13): 2.6, 4.5, 4.6 (n=2: just do these problems for the case of two factors.)
6.10 (b), "only if" part ("if" part already  done in class.)

9/9 M Problem 6.10 (c)/ Existence of metric completion via equivalence classes of Cauchy seqs. (start)

9/11 W Metric completion (cont'd)--uniqueness statement.

9/13 F Uniform continuity. Examples (Lipschitz and Hölder conditions), extension of unif cont maps(with values in complete spaces) to the closure of a set.

9/16 M Completion via distance function; uniqueness (see handout)
Metric completion via distance functions
(includes homework problems. Problems 2,3,4=Problem set 4, due 9/20)

9/18 W Completeness of the real line
(completeness for absolute value + Archimedean property) equivalent to supremum property

Two Topological Uniqueness Theorems for Spaces of Real Numbers by M.Francis (ArXiv link.)
(read as the course progresses, with a view to possible presentation topic.)

9/20 F Finite-dimensional normed linear spaces.
Equivalence of norms/norms defined by balanced convex sets
Equivalence of norms
Problem set 5 (due 9/27) from text, chapter One: 3.1(a)(b)7.6 (c), 7.7(b)(c) (using (a))
Hw 5 Notes

9/23 M Compactness: metric spaces (Sect 1.5)

9/25 W Cantor sets

9/27 F  Compactness in topological spaces (start); Heine-Borel for general metric spaces. (sect 1.5)
Problem set 6 (due 10/4) 1.5.4(a), 1.5.7, 1.5.8(only the "rho is a metric" part), 2.6.2.
(Think about 1.5.1, 1.5.2. for discussion.)

9/30 M Compactness in topological spaces:
base of a topology, second countable spaces, Lindeloef's theorem.(sect 1.5)
compactness, continuity and homeomorphisms.(sect 2.6)

10/2  W Compactness vs. sequential compactness, countability conditions and separability,
Example: C(R,[0,1]) is non-separable metric (and complete.)

10/4 F Ex: cont functions on compact metric spaces are unif. cont. Defining topologies: base and subbase. Examples (product topology, Sorgenfrey line, removing 1/n). (2.4)
Problem set 7 (due 10/11) 2.4.6, 2.5.6

10/7 M  Extension of continuous functions :normal spaces, Urysohn's lemma

10/9 W Normal spaces and regular spaces; second countable+regular implies normal
Normal spaces and Urysohn metrization
(summary--includes four problems=Problem set 8, due 10/25.)

10/11 F l^2 and the Hilbert cube

10/14 M Hilbert cube, Urysohn metrization

10/16 W lecture postponed

10/18 F Fall Break (no lecture)

10/21 M Connected spaces

10/22 T (makeup: A404, 10:00-10:50): path-connected spaces

10/23 W locally path conected /locally connected spaces
Problem set 9: (2.8) 7, 8, 9, 10, 11

10/25 F   Arzela-Ascoli theorem (start--equicontinuity, main lemmas)

10/28 M Arzela-Ascoli (end of proof, extension.)

10/30 W example from calc of variations/Baire's theorem (start)

11/1  F Baire's theorem (proof); examples (stabiltiy and genericity)
HW set 10: problems 1, 3, 4, 10 (p. 7/8 of Arzela-Ascoli notes.) Due 11/8
HW10 Notes

11/4  M Remarks on Baire's theorem and G-delta sets
Remarks on Baire Spaces and G-delta sets

11/6  W  Uniform boundedness/fixed point for contractions

11/8  F  Picard's theorem/topological vector spaces (start)
TVS basics
HW set 11 (due 11/15): 1.8 (p. 45 in text): 3, 4, 6.

11/11 M TVS basics: metrization

11/13 W  Locally compact Banach spaces are finite dimensional
Locally Compact Banach spaces are finite-dimensional

11/15 F   Characterization of Cantor set (Matthew)
Topological Characterization of Cantor Sets
(presentation by Matthew Shaw)
Problem set 12
(due 11/22)

11/18 M  TVS basics: seminorms, Kolmogorov's normalization theorem

11/20 W Stone-Weierstrass Theorem
(presentation by Ethan Kessinger)

11/22 F  Nowhere differentiable continuous functions are generic
(presentation by Billy Reynolds)

11/25 M Product topology and Tychonoff's theorem
Products, Tychonoff's theorem, compactifications
HW set 13: the four problems in this handout (due 12/10)

11/27 W No lecture

11/29 F Thanksgiving break

12/2  M Compactifications

12/4 W Topological characterization of the interval and the circle
(presentation by Jacob Honeycutt and Shane Butler)

12/10 T  (day scheduled for final)--turn in HW set 13.