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
(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.