1 Basics of Topology

“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”
– John von Neumann

– John von Neumann was a mathematician with groundbreaking contrubutions to mathematics and physics.

We will start with some very basic concepts of metric topology and work through some problems to familiarize ourselves with these concepts. For more on these topics, you can refer to any textbook on metric topology. Possible references more geared to our immediate needs for this course are

1.1 Metric Spaces

Metric on a Set

Let M set of points. A function d: M \times M \to \mathbb{R} is said to be a metric if the following conditions are satisfied:

For each a,b \in M, d(a,b) \ge 0
For each a,b \in M , d(a,b) = 0 \iff a=b
For each a,b \in M, d(a,b)=d(b,a)
For each a,b,c \in M, d(a,c) \le d(a,b) + d(b,c)

Exercise 1.1

(familiar example) The set \mathbb{R} with d(a,b)=|a-b|
(familiar example) The set \mathbb{R}^2 with d_2((x,y),(x',y'))=\sqrt{(x-x')^2+ (y-y')^2}
The set \mathbb{R}^2 with d_{\infty}((x,y),(x',y'))=\max\{{|x-x'|, |y-y'|}\}
The set \mathbb{R}^2 with d_1((x,y),(x',y'))=|x-x'| + |y-y'|
on any non-empty set M, set d(a,a)=0, and d(a,b)=1 whenever a \ne b
If f,g : [0,1] \to \mathbb{R} be continuous functions, define d(f,g)= \int_0^1 |f(x)-g(x)| dx. Then d is a metric on the set of all continuous real valued functions on [0,1]
Let (M,d) be a metric space. Show that for x,y,z \in M we have |d(x,z)-d(y,z)|\le d(x,y)
Let (X,d) be a metric space. Are the following functions d: X \times X \to \mathbb{R} metrics on X?

d(x,x')=\sqrt{d(x,x')}
d(x,x')={[d(x,x')]}^2

For any real numbers a,b,c,d show that (ab+cd)^2 \le (a^2+c^2)(b^2+d^2)

Open Sets and closed sets; limit points

Let (M,d) be a metric space, a \in M and r>0. We define the open ball centered at a with radius r as B(a,r):=\{x \in M: d(x,a) <r\}, and the closed ball centered at a with radius r as \bar{B}(a,r):=\{x \in M: d(x,a) \le r\}
A subset U \subset M is said to be open in M iff for x \in U, there is r>0 such that B(x,r) \subset U
A subset C \subset M is said to be closed in M iff M\setminus C is open in M
Let (M,d) be a metric space, A \subset M. A point x \in M is said to be a limit point of A iff each open ball centered at x contains at least one point of A\setminus \{x\}

Exercise 1.2

Revisit the examples of metric spaces above and try to describe what the open balls in each case are (give geometric descriptions - with sketches - whenever possible)
Show that open balls in a metric space are themselves open sets
Show that, in a metric space M, the empty set \emptyset is open, as is the entire space M.
If subsets U_1, U_2 are both open so are U_1 \cup U_2 and U_1 \cap U_2
Consider a finite number of open subsets U_1, \cdots, U_n of a metric space. Are U_1 \cup \cdots \cup U_n and U_1 \cap \cdots \cap U_n open?
Consider possibly infinite number of open subsets U_1, \cdots, U_i, \cdots of a metric space. Are U_1 \cup \cdots \cup U_i \cup \cdots and U_1 \cap \cdots \cap U_i \cap \cdots open?
Show that, in a metric space, a subset is open iff it is the union of open balls.
Show that closed balls in a metric space are themselves closed sets
Show that, in a metric space M, the empty set \emptyset is closed, as is the entire space M.
If subsets C_1, C_2 are both closed so are C_1 \cup C_2 and C_1 \cap C_2
Consider a finite number of closed subsets C_1, \cdots, C_n of a metric space. Are C_1 \cup \cdots \cup C_n and C_1 \cap \cdots \cap C_n closed?
Consider possibly infinite number of closed subsets C_1, \cdots, C_i, \cdots of a metric space. Are C_1 \cup \cdots \cup C_i \cup \cdots and C_1 \cap \cdots \cap C_i \cap \cdots closed?
Show that, in a metric space, a subset is closed iff it contains all its limit points.

1.2 Continuous functions between metric spaces

Continuous (and uniformly continuous) functions

Let (X,d_X) and (Y,d_Y) be metric spaces and f:X \to Y be a function. f is said to be continuous at a \in X iff for every \epsilon >0 there is \delta > 0 such that for every x \in X such that d_X(a,x) < \delta we have d_Y(f(x),f(a))<\epsilon. If f is continuous at every a \in X we say that f is continuous.
Let (X,d_X) and (Y,d_Y) be metric spaces and f:X \to Y be a function. f is said to be uniformly continuous iff for every \epsilon >0 there is \delta > 0 such that for every a,x \in X with d_X(a,x) < \delta we have d_Y(f(x),f(a))<\epsilon.

Exercise 1.3

(familiar examples) All the functions f:\mathbb{R} \to \mathbb{R} you use in first year calculus are continuous in their domain. Try to prove the continuity of some of them using the above definition
Show that a constant function from any metric space to another is continuous.
Give an example of a continuous function f:\mathbb{R} \to \mathbb{R} that is not uniformly continuous
Give an example of a continuous function f:(0,1) \to \mathbb{R} that is not uniformly continuous
(distance from a fixed point) Let (X,d) be a metric space and a \in X. Show that the function f_a: X \to \mathbb{R} given by f_a(x)=d(x,a) is continuous.
Let M be the metric space defined in Exercise 1.1.5. Show that any function from M to any metric space must be continuous
Suppose we have a continuous function f:X \to Y with an inverse function f^{-1}: Y \to X. Must f^{-1} be continuous?
Show that if f,g : (X,d) \to {\mathbb{R}} are both continuous, then so are f+g, cf and fg. [note that convergent sequences of real numbers are bounded]
Let X,Y,Z be metric spaces. If f: X \to Y and g: Y \to Z are continuous, show that g \circ f : X \to Z is continuous.
Prove that if f:X \to Y satisfies d_Y(f(x),f(y)) \le Cd_X(x,y) for some constant C > 0, then f is continuous.

Sequential Characterization of continuity

Definition: Let (X,d_X) be a metric space. A sequence \{x_n\} \subset X is said to converge to a \in X iff for every \epsilon > 0 there is a positive integer N \in \mathbb{N} such that n \ge N \implies d_X(x_n,a) < \epsilon.

Proposition: Let f:(X,d_x) \to (Y,d_Y) be a function and let a \in X. TFAE

f is continuous at a
For every converging sequence \{x_n\} \subset X converging to a, the sequence \{f(x_n)\} converges to f(a) in Y

Isometries

Definition: A function f:(X,d_X) \to (Y,d_Y) is an isometry iff it is bijective and for every x,x' \in X, we have d_X(x,x')=d_Y(f(x),f(x'))

Exercise 1.4

What are the isometries of \mathbb{R}?
What are the isometries of the Euclidean plane?

1.3 Homeomorphisms

Homeomorphisms

Definition: A function f:(X,d_X) \to (Y,d_Y) is an homeomorphism iff

it is bijective
f is continuous
f^{-1} (the inverse function of f) is continuous

f:X \to Y is one-to-one (injective) iff f(x_1)=f(x_2) implies x_1=x_2
f: X \to Y is onto (surjective) iff for each y \in Y there is x \in X such that f(x)=y
f: X \to Y is bijective iff it is BOTH one-to-one and onto

Exercise 1.5

Give an example of a bijective function from one metric space to another which is not continuous.
Give an example of a continuous function from one metric space to another which is not bijective.
Give an example of a bijective continuous function from one metric space to another, such that the inverse function f^{-1} is not continuous.
Show that \mathbb{R} \cong (-\pi/2, \pi/2)
Show that (-\pi/2, \pi/2) \cong (0,1)
Explain why does the above imply that \mathbb{R} \cong (0,1)
Show that the open unit disk on the Euclidean plane is homeomorphic to \mathbb{R}^2
Show that, as subspaces of \mathbb{R}^2, the (boundary of the) unit square is homeomorphic to the (the boundary of the) unit disk
Show that (0,1) \not\cong [0,1]
Let X and Y be metric spaces, each with finitely many points. Show that X \cong Y iff X and Y have the same number of points.

1.4 Homotopy

Homotopy between continuous functions

Two functions f,g:(X,d_X) \to (Y,d_Y) are said to be homotopic (f \sim g) iff there is a continuous function F: X \times [0,1] such that F(x,0)= f(x) and F(x,1)= g(x) for all x \in X.
A continuous function f: X \to Y is said to be a homotopy equivalence iff there is a continuous function g: Y \to X such that g \circ f \sim \textbf{id}_X and f \circ g \sim \textbf{id}_Y.
Spaces X,Y are said to be homotopy equivalent iff there is a homotopy equivalence between them.
A space is said to be contractible iff the identity function is homotopic to a constant function (the function that takes the entire space to a single point in the space).

Exercise 1.6

Let f, g: \mathbb{R}^2 \to \mathbb{R}^2 be defined as f(\bar{x})=\bar{x} and g(\bar{x})=\bar{0} for \bar{x} \in \mathbb{R}^2. Show that f \sim g.
Let f, g: X \to Y be continuous functions from any metric space R^n to a convex subspace of \mathbb{R}^n. Show that f \sim g.
Let f,g:X \to Y be continuous. Define a (continuous by definition) function F: (X \times \{0\}) \cup (X \times \{1\}) by F(x,0)=f(x) and F(x,1)=g(x). Show that f \sim g iff there is a continuous extension of F to X \times [0,1].
Show that homotopy of continuous functions from a space X to a space Y is an equivalence relation.
Show that a homeomorphism between spaces X and Y is a homotopy equivalence, but not conversely.
Show that \mathbb{R}^2\setminus\{(0,0)\} \sim S^1.
Show that any convex subset of \mathbb{R}^n is contractible.

1.5 Distance between subsets of a metric space

Distance between finite subsets

Let (X,d) be a metric space and A, B be finite subsets of X. Then the distance between A and B (in the metric d) is defined as d(A,B)= \min \{d(a,b): a \in A, b\in B\}
Let (X,d) be a metric space and A, B be finite subsets of X. Then the Directed Hausdorff distance from A to B is defined as \overrightarrow{d}_{H}(A,B)= \max_{a \in A} d(a,B)=\max_{a \in A} \min_{b \in B}(a,b)
Let (X,d) be a metric space and A, B be finite subsets of X. Then the Hausdorff distance between A and B is defined as \begin{align*} d_H(A,B)& = \max \{\overrightarrow{d}_{H}(A,B), \overrightarrow{d}_{H}(B,A)\} \\ & = \max \{\max_{a \in A} d(a,B), \max_{b \in B} d(b,A) \} \end{align*}

If the subsets A, B are not finite, the Hausdorff distance between them is d_H(A,B)= \max \{\sup_{a \in A} d(a,B), \sup_{b \in B} d(b,A) \}

Exercise 1.7

Find the Euclidean distance between the subsets A=\{(1,0),(2,0),(3,0)\} and B=\{(1,1),(2,2),(3,3)\} of \mathbb{R}^2
Find the Hausdorff distance between the subsets A=\{(1,0),(2,0),(3,0)\} and B=\{(1,1),(2,2),(3,3)\} of \mathbb{R}^2
Find the Euclidean distance between the subsets A=\{(x,0): -\infty < x< \infty\} and B=\{(x,\frac{1}{1+x^2}): -\infty < x< \infty\} of \mathbb{R}^2
Find the Hausdorff distance between the subsets A=\{(x,0): -\infty < x< \infty\} and B=\{(x,\frac{1}{1+x^2}): -\infty < x< \infty\} of \mathbb{R}^2
Show that the Hausdorff distance between finite subsets A, B of a metric space (X,d) can be expressed as d_H(A,B)= \min \{\epsilon > 0: A \subset \mathcal{N}_{\epsilon}(B) \text{ and } B \subset \mathcal{N}_{\epsilon}(A) \} where \mathcal{N}_{\epsilon}(A) is the set of all points of X at distance less than \epsilon from A
Let \mathcal{F} be the set of all finite subsets of a metric space (X,d). Show that the Hausdorff distance makes \mathcal{F} into a metric space.

If the subsets A, B are not finite, the Hausdorff distance between them is d_H(A,B)= \inf \{\epsilon > 0: A \subset \mathcal{N}_{\epsilon}(B) \text{ and } B \subset \mathcal{N}_{\epsilon}(A) \}

where \mathcal{N}_{\epsilon}(A) is the set of all points of X at distance less than \epsilon from A