MATH20122: Starter Problems
Starter Problem - Lecture 11.
Let S R2 denote the circle with centre (0, 0) and radius 1, equipped with
the arc-length metric dS of Problem 3; prove that e(x) = (cos x, sin x) defines a
continuous function e : R S from the Eucl
Chapter 5
Compactness
Lecture 14
As in previous sections, (X, d) is a metric space and A X an arbitrary subset.
Definitions 5.1. A covering of A is a collection of sets U = cfw_Ui : i I for
which
[
A
Ui ;
iI
a subcovering of U is a subcollection cfw_Ui :
Chapter 4
Continuous Functions
Lecture 11
Definitions 4.1. Given metric spaces (X, dX ) and (Y, dY ), a function f : X Y
is continuous at x0 in X whenever
> 0, > 0 such that dX (x, x0 ) < dY (f (x), f (x0 ) < .
If f is continuous at every x0 X, then f is
Chapter 1
Definitions and Examples
Lecture 1
Definition 1.1. A metric space (X, d) consists of a non-empty set X and a
non-negative real valued metric d : X X R> which satisifes the axioms
1. d(x, y) = 0 x = y for all x, y X
2. d(x, y) = d(y, x) for all x
Chapter 3
Uniform Convergence
Lecture 9
Sequences of functions are of great importance in many areas of pure and applied
mathematics, and their properties can often be studied in the context of metric
spaces, as in Examples 2.17.4 and .5.
Unless otherwise
Chapter 6
Completeness
Lecture 18
Recall from Definition 2.22 that a Cauchy sequence in (X, d) is a sequence whose
terms get closer and closer together, without any limit being specified. In the
Euclidean line, every Cauchy sequence converges; but there a
Starter Problems 5
Starter Problem - Lecture 14.
Describe an open covering U = cfw_Ui : i I (such that I is infinite) of the
square cfw_x : 0 x1 , x2 1 in the Euclidean plane. Find a finite subcovering.
Starter Problem - Lecture 15.
Construct a non-conver
Starter Problem - Lecture 1.
In any metric space (X, d), prove that
B2 (x) B3 (x), B 2 (x) B3 (x), and B 2 (x) B 3 (x)
for any x X.
Starter Problem - Lecture 2.
Suppose the graph lies in the complex plane, with 13 vertices
V = cfw_0, 1, 2, 3, i, 2i, 3i
an
MATH20122: Starter Problems
Starter Problem - Lecture 7.
Consider the graph with 5 vertices cfw_0, 1, i and 4 edges cfw_[0, i], [0, 1]
in the complex plane; with respect to the edge metric e on V , give examples of a
non-convergent sequence (un ), and a n
Chapter 2
Open and Closed Sets
Lecture 5
Given any metric space (X, d), it is extremely important to generalise the notion
of an open ball. So throughout this section, let U X be an arbitrary subset.
Definition 2.1. An interior point u U is one for which