Homework 4
Math 501
Due September 26, 2014
Throughout this assignment, unless otherwise stated, we think of Rm as a
metric space with the usual Euclidean metric induced by the Euclidean norm.
In general, when thinking of a metric space as a topological space, it is implied
that we use the metric topology.
Exercise 1
Consider the following sets:
A1 = (0, 1),
A2 = {(x, 0) ∈ R2 : x ∈ (0, 1)}.
In a sense, these two sets are same; A2 is simply the embedding of A1 into R2 .
However, we see below that this small distinction makes a difference topologically.
(a) Briefly explain why A1 is open in R.
(b) Explain why A2 is not open in R2 .
Exercise 2
Let M be a metric space and suppose that S ⊂ T ⊂ M . Prove that
(a) S ⊂ T ,
(b) int S ⊂ int T .
Exercise 3
Let X be a topological space. A subset U ⊂ X is said to be dense in X if
U = X.
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Let M be a metric space and let A ⊂ B ⊂ C ⊂ M . Suppose that A is dense
in B and B is dense in C. Prove that A is dense in C.
Exercise 4
If M is a metric space, any subset S ⊂ M of M is itself naturally a metric
space by restricting the metric on M to S. For instance, the set N ⊂ R is a
metric space with the usual Euclidean metric d(n, m) = |n − m| inherited from
R.
(a) Show that every subset of N is both closed and open in the metric topology
on N inherited from R. [Sets that are both open and closed are often referred
to as clopen.]
(b) Show that any function f : N → M is continuous, whenever M is a metric
space.
Exercise 5
The distance from a point p ∈ M in a metric space to a non-empty subset
S ⊂ M is defined to be
dist(p, S) = inf{d(p, s) : s ∈ S}.
Show that p is a limit point of S if and only if dist(p, S) = 0.
Exercise 6
Consider a sequence (xn ) ⊂ R that converges. Prove that the sequence (|xn |)
of absolute values converges. Prove or disprove the converse.
Exercise 7
If X and Y are topological spaces and f : X → Y is a function, we denote
the image of X under f by the set
f (X) = {f (x) ∈ Y : x ∈ X}.
The map f : X → Y is said to be an embedding if it is a homeomorphism
onto its image. That is, the map f : X → f (X) is a homeomorphism. If there
exists such an embedding, we say that X embeds in Y .
(a) Find an embedding f : R → R2 such that f (R) is a closed subset of R2 .
(b) A set A ⊂ R2 is said to be bounded if there exists M > 0 such that
A ⊂ BM (0). For example, the unit circle S 1 = {x ∈ R2 : |x| = 1} is a
2
bounded set because it fits inside the ball of radius 2. We aim to show
below that there exists an embedding g : R → R2 such that g(R) is a
bounded set:
Let N = (0, 1) ∈ R2 denote the so-called “north-pole” of S 1 . Consider the
map g : R → S 1 \ {N } which is defined as follows. For each x ∈ R, consider
the line `x ⊂ R2 which connects (x, 0) and N . Define g(x) ∈ R2 to be the
point of intersection of `x with S 1 \ {N }.
(i) Find a formula for g(x).
(ii) Argue that g is a homeomorphism onto its image.
(c) The two parts above show, respectively, that R embeds in R2 as a closed
subset and as a bounded subset. Explain why there cannot exist an embedding h : R → R2 such that h(R) is both closed and bounded.
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