Topology - Homework assignment 1
[Due Wednesday, February 11th]
Instructions: If you are taking the course as Math 440, you only need to complete
problems 1 3. If you are enrolled in the course as Math 640, you need to turn in all 4
problems.
1. Given a n
Topology - Homework assignment 3
[Due Monday, March 2nd]
Instructions: If you are taking the course as Math 440, you only need to complete
problems 1 2. If you are enrolled in the course as Math 640, you need to turn in all 3
problems.
1. Consider the seq
Topology - Homework assignment 2
[Due Wednesday, February 18th]
Instructions: If you are taking the course as Math 440, you only need to complete
problems 1 3. If you are enrolled in the course as Math 640, you need to turn in all 4
problems.
1. Let X be
Topology - Homework assignment 4
[Due Wednesday, April 1st]
Instructions: If you are taking the course as Math 440, you only need to complete
problems 1 3. If you are enrolled in the course as Math 640, you need to turn in all 4
problems.
1. Let (X, TX )
CHAPTER 1
Continuity and convergence in Euclidean spaces
This section serves as motivation for the denitions of a topological space X and a
continuous function f : X Y between two topological spaces X and Y , both given
in section ?. These denitions are s
CHAPTER 1
Separation axioms
1. Degrees of separation
In chapter ? we already encountered two measures for the size of topology T on a
set X , namely those of being separable (denition ?) and of being second countable
(denition ?). These two measures howev
CHAPTER 1
Continuous functions and convergent sequences
1. Continuous functions
Definition 1.1. Let (X, TX ) and (Y, TY ) be two topological spaces and let f :
X Y be a function.
(a) We say that f is continuous at x X if for every neighborhood V of f (x)
CHAPTER 1
Topological spaces
1. Denition of a topological space
Definition 1.1. A topology T on a set X is a collection of subsets of X subject to
the following three rules, called the axioms of a topology:
1. The empty set and all of X belong to T .
2. I