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

(presentation by Mohammad Islam)

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.