Contents of this page:
Part of the linear algebra program at the Nesin Mathematics Village, January 16–22, 2017.
Lecture notes, last edited January 5, 2018: 24 pages, size A5, 12 point type.
See also last year's course.
The Cantor Set // Cantor Kümesi
A course at the Nesin Mathematics Village, January 23–9, 2017.
Lecture notes, edited January 11, 2018: 38 pages, size A5, 12 point type.
Georg Cantor (1845–1918) proved the theorem called by his name, that the collection of subsets of a given set is strictly larger than the set itself. There is an embedding of the set in the collection of its subsets; but the range of the embedding cannot be the whole collection of subsets. In particular, while the set of natural numbers is (by definition) countable, the collection of its subsets is uncountable. Cantor showed also that the set of real numbers is uncountable. In fact it has the same size as the collection of sets of natural numbers; and one proof of this result uses what we now call the Cantor set.
An open set of real numbers is a union of countably many open intervals. The complement of an open set of real numbers is a closed set. The Cantor set is a counterexample to the conjecture that every closed set of real numbers is a countable union of closed intervals and one-element sets.
We shall prove these results carefully and, as time and interest permits, consider uses, appearances, and generalizations of the Cantor set in analysis, topology, algebra, and logic.
A course at the Nesin Mathematics Village, July 31–August 6, 2017.
A course at the Nesin Mathematics Village, August 7–13, 2017.
Notes in rough form (67 pages, size A5, 12 point type):