Metric Spaces

Let $S$ be a subset of $\mathbb {R}$. Let $C$ be the set of points $x$ in $\mathbb {R}$ with the property that $S \cap (x-\delta , x+\delta )$ is uncountable for every $\delta > 0$. Prove that $S \setminus C$ is finite or countable.

Let $A \subseteq \mathbb {R}$ be uncountable.

Let $\left\{ x(i,j) \mid i, j \in \mathbb {N}\right\}$ be a doubly indexed set in a complete metric space $(X, \rho )$ such that

Prove that the iterated limits $\lim_{i \to \infty } \lim_{j \to \infty } x(i,j)$, $\lim_{j \to \infty } \lim_{i \to \infty } x(i,j)$ exist and are equal.

Let $T_0$ be the interior of a triangle in $\mathbb {R}^{2}$ with vertices $A, B, C$. Let $T_1$ be the interior of the triangle whose vertices are the midpoints of the sides of $T_0$, $T_2$ the interior of the triangle whose vertices are the midpoints of the sides of $T_1$, and so on. Describe the set $\cap_{n=0}^{\infty } T_n$.

Let $K$ be a compact subset of $\mathbb {R}^{n}$ and $\left\{ B_j\right\}$ a sequence of open balls that covers $K$. Prove that there is a positive number $\varepsilon$ such that each $\varepsilon$-ball centered at a point of $K$ is contained in one of the balls $B_j$.

Prove or disprove: The set $\mathbb {Q}$ of rational numbers is the intersection of a countable family of open subsets of $\mathbb {R}$.

Let $X \subset \mathbb {R}$ be a nonempty connected set of real numbers. If every element of $X$ is rational, prove $X$ has only one element.

Give an example of a subset of $\mathbb {R}$ having uncountably many connected components. Can such a subset be open? Closed?

Let $f_n : \mathbb {R}^{k} \to \mathbb {R}^{m}$ be continuous ($n = 1, 2, \ldots$). Let $K$ be a compact subset of $\mathbb {R}^{k}$. Suppose $f_n \to f$ uniformly on $K$. Prove that

is compact.

Show that the interval $[0,1]$ cannot be written as a countably infinite disjoint union of closed subintervals of $[0,1]$.

Let $X$ and $Y$ be nonempty subsets of a metric space $M$. Define

Let $S \subset \mathbb {R}^{n}$ be a subset which is uncountable. Prove that there is a sequence of distinct points in $S$ converging to a point of $S$.

Let $X \subset \mathbb {R}^{n}$ be a closed set and $r$ a fixed positive real number. Let $Y = \left\{ y \in \mathbb {R}^{n} \mid \left|x-y\right| = r \text{ for some } x \in X\right\}$. Show that $Y$ is closed.

Show that every infinite closed subset of $\mathbb {R}^{n}$ is the closure of a countable set.

Let $\left\{ U_1, U_2, \ldots \right\}$ be a cover of $\mathbb {R}^{n}$ by open sets. Prove that there is a cover $\left\{ V_1, V_2, \ldots \right\}$ such that

A standard theorem states that a continuous real valued function on a compact set is bounded. Prove the converse: If $K$ is a subset of $\mathbb {R}^{n}$ and if every continuous real valued function on $K$ is bounded, then $K$ is compact.

Let $A \subset \mathbb {R}^{n}$ be compact, $x \in A$; let $(x_i)$ be a sequence in $A$ such that every convergent subsequence of $(x_i)$ converges to $x$.

Let $X \subset \mathbb {R}^{n}$ be compact and let $f : X \to \mathbb {R}$ be continuous. Given $\varepsilon > 0$, show there is an $M$ such that for all $x, y \in X$,

Let $\left\{ S_{\alpha }\right\}$ be a family of connected subsets of $\mathbb {R}^{2}$ all containing the origin. Prove that $\cup_{\alpha } S_{\alpha }$ is connected.

Consider the following properties of a function $f : \mathbb {R}^{n} \to \mathbb {R}$:

1. $f$ is continuous.

2. The graph of $f$ is connected in $\mathbb {R}^{n} \times \mathbb {R}$. Prove or disprove the implications $1 \Rightarrow 2$, $2 \Rightarrow 1$.

Let $U$ be a nonempty, proper, open subset of $\mathbb {R}^{n}$. Construct a function $f : \mathbb {R}^{n} \to \mathbb {R}$ that is discontinuous at each point of $U$ and continuous at each point of $\mathbb {R}^{n} \setminus U$.

Prove or give a counterexample: Every connected, locally pathwise connected set in $\mathbb {R}^{n}$ is pathwise connected.

The set of real $3 \times 3$ symmetric matrices is a real, finite-dimensional vector space isomorphic to $\mathbb {R}^{6}$. Show that the subset of such matrices of signature $(2,1)$ is an open connected subspace in the usual topology on $\mathbb {R}^{6}$.

Let $M_{n \times n}$ be the vector space of real $n \times n$ matrices, identified with $\mathbb {R}^{n^2}$. Let $X \subset M_{n \times n}$ be a compact set. Let $S \subset \mathbb {C}$ be the set of all numbers that are eigenvalues of at least one element of $X$. Prove that $S$ is compact.

Let $SO(3)$ denote the group of orthogonal transformations of $\mathbb {R}^{3}$ of determinant $1$. Let $Q \subset SO(3)$ be the subset of symmetric transformations $\neq I$. Let $P^2$ denote the space of lines through the origin in $\mathbb {R}^{3}$.

Let $m$ and $n$ be positive integers, with $m < n$. Let $M_{m \times n}$ be the space of linear transformations of $\mathbb {R}^{m}$ into $\mathbb {R}^{n}$ (considered as $n \times m$ matrices) and let $L$ be the set of transformations in $M_{m \times n}$ which have rank $m$.

Let $M_{n \times n}$ be the space of real $n \times n$ matrices. Regard it as a metric space with the distance function

Prove that the set of nilpotent matrices in $M_{n \times n}$ is a closed set.

Let $S$ be an uncountable subset of $\mathbb {R}$. Prove that there exists a real number $t$ such that both sets $S \cap (-\infty , t)$ and $S \cap (t, \infty )$ are uncountable.

Let $(X, \rho )$ and $(Y, \sigma )$ be metric spaces. Assume that:

1. $f, f_1, f_2, \ldots$ are bijective functions of $X$ onto $Y$ with inverses $g, g_1, g_2, \ldots$

2. $g$ is uniformly continuous

3. $f_n \to f$ uniformly as $n \to \infty$. Prove that $g_n \to g$ uniformly as $n \to \infty$.

Let $E_1, E_2, \ldots$ be nonempty closed subsets of a complete metric space $(X,d)$ with $E_{n+1} \subset E_n$ for all positive integers $n$, and such that $\lim_{n \to \infty } \operatorname {diam}(E_n) = 0$, where $\operatorname {diam}(E)$ is defined to be

Prove that $\bigcap_{n=1}^{\infty } E_n \neq \emptyset$.

Let $A$ be a subset of a compact metric space $(X,d)$. Assume that, for every continuous function $f : X \to \mathbb {R}$, the restriction of $f$ to $A$ attains a maximum on $A$. Prove that $A$ is compact.

Let $X$ be a metric space and $(x_n)$ a convergent sequence in $X$ with limit $x_0$. Prove that the set $C = \left\{ x_0, x_1, x_2, \ldots \right\}$ is compact.

Prove that every compact metric space has a countable dense subset.

Let $X$ be a compact metric space and $f : X \to X$ an isometry. Show that $f(X) = X$.

Let $M$ be a metric space with metric $d$. Let $C$ be a nonempty closed subset of $M$. Define $f : M \to \mathbb {R}$ by

Show that $f$ is continuous, and that $f(x) = 0$ if and only if $x \in C$.

Let $C^{1/3}$ be the set of real valued functions $f$ on the closed interval $[0,1]$ such that

Let $\left\{ f_n\right\}_{n=1}^{\infty }$ be a uniformly bounded equicontinuous sequence of real-valued functions on the compact metric space $(X,d)$. Define the functions $g_n : X \to \mathbb {R}$, for $n \in \mathbb {N}$ by

Prove that the sequence $\left\{ g_n\right\}_{n=1}^{\infty }$ converges uniformly.

Let $\mathcal{F}$ be a uniformly bounded, equicontinuous family of real valued functions on the metric space $(X,d)$. Prove that the function

is continuous.

Let $X$ and $Y$ be metric spaces and $f$ a continuous map of $X$ into $Y$. Let $K_1, K_2, \ldots$ be nonempty compact subsets of $X$ such that $K_{n+1} \subset K_n$ for all $n$, and let $K = \bigcap K_n$. Prove that $f(K) = \bigcap f(K_n)$.

Let $(X_1, d_1)$ and $(X_2, d_2)$ be metric spaces and $f : X_1 \to X_2$ a continuous surjective map such that $d_1(p,q) \leq d_2(f(p), f(q))$ for every pair of points $p, q$ in $X_1$.

An accurate map of California is spread out flat on a table in Evans Hall, in Berkeley. Prove that there is exactly one point on the map lying directly over the point it represents.

Define a sequence of positive numbers as follows. Let $x_0 > 0$ be any positive number, and let $x_{n+1} = (1+x_n)^{-1}$. Prove that this sequence converges, and find its limit.

Let $f : \mathbb {R} \to \mathbb {R}$ be monotonically increasing (perhaps discontinuous). Suppose $0 < f(0)$ and $f(100) < 100$. Prove $f(x) = x$ for some $x$.

Let $K$ be a nonempty compact set in a metric space with distance function $d$. Suppose that $\varphi : K \to K$ satisfies

for all $x \neq y$ in $K$. Show there exists precisely one point $x \in K$ such that $x = \varphi (x)$.

Let $K$ be a continuous function on the unit square $0 \leq x, y \leq 1$ satisfying $\left|K(x,y)\right| < 1$ for all $x$ and $y$. Show that there is a continuous function $f(x)$ on $[0,1]$ such that we have

Can there be more than one such function $f$?

Let $g$ be a continuous real valued function on $[0,1]$. Prove that there exists a continuous real valued function $f$ on $[0,1]$ satisfying the equation

Show there is a unique continuous real valued function $f : [0,1] \to \mathbb {R}$ such that

Let $(M,d)$ be a nonempty complete metric space. Let $S$ map $M$ into $M$, and write $S^2$ for $S \circ S$; that is, $S^2(x) = S(S(x))$. Suppose that $S^2$ is a strict contraction; that is, there is a constant $\lambda < 1$ such that for all points $x, y \in M$, $d(S^2(x), S^2(y)) \leq \lambda d(x,y)$. Show that $S$ has a unique fixed point in $M$.