Topology of $\reals[n]$

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.