Section 31: Problem 9 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

$^{*}$ 9. Let $A$ be the set of all points of $\mathbb{R}_{l}^{2}$ of the form $x\times(—x)$ , for $x$ rational; let $В$ be the set of all points of this form for $x$ irrational. If $V$ is an open set of $\mathbb{R}_{l}^{2}$ containing $B$ , show there exists no open set $U$ containing $A$ that is disjoint from $V$ , as follows:

(a) Let $K_{n}$ consist of all irrational numbers $x$ in $[0,1]$ such that $[x,x+1/n)\times[-x,-x+1/n)$ is contained in $V$ . Show $[0,1]$ is the union of the sets $K_{n}$ and countably many one-point sets.

(b) Use Exercise 5 of §27 to show that some set $\overline{K}_{n}$ contains an open interval $(a,b)$ of $\mathbb{R}$ .

(c) Show that $V$ contains the open parallelogram consisting of all points of the form $x\times(-x+\epsilon)$ for which $a<x<b$ and $0<\epsilon<1/n$ .

(d) Conclude that if $q$ is a rational number with $a<q<b$ , then the point $q\times(-q)$ of $\mathbb{R}_{l}^{2}$ is a limit point of $V$ .

(a) For every irrational point $x\in[0,1]$ there is a basis neighborhood $[x,x+\epsilon_{x})\times[-x,-x+\delta_{x})$ contained in $V$ , therefore, if $n>1/\min\{\epsilon_{x},\delta_{x}\}$ then $x\in K_{n}$ . The rest of $[0,1]$ , i.e. the rational points, is a countable set of points.

(b) According to the result of the Exercise 5 of §27, since $\cup_{n}\overline{K}_{n}\cup\cup_{q\in[0,1]\cap\mathbb{Q}}\{q\}=[0,1]$ has non-empty interior in $[0,1]\subset\mathbb{R}$ being compact and Hausdorff, one of the closed sets in the union must have a non-empty interior in $[0,1]\subset\mathbb{R}$ as well. Therefore, there is $n$ such that $\overline{K}_{n}$ has non-empty interior, and contains a non-empty interval $(a,b)$ .

(c) I show a little different result: for $n$ found in (b) there is a non-empty interval $(a',b')\subset[0,1]$ and $0<\epsilon<1/n$ such that for every $x\in(a',b')$ , $x\times(-x,-x+\epsilon)\subseteq V$ . Take $\epsilon<\min\{1/n,(b-a)/2\}$ where $n$ and $a<b$ are such that $(a,b)\subset\overline{K}_{n}$ (found in (b)). Let $A=(a',b')=(a+\epsilon,b-\epsilon)$ . $A$ is not empty. For each $x\in A$ and $0<\delta<\epsilon$ there is a point $z\in(x-\delta,x)\cap K_{n}$ . Therefore, $(x,-x+\delta)=(z+x-z,-z+\delta-(x-z))\in[z,z+1/n)\times[-z,-z+1/n)\subset V$ .

(d) Given (c), for every point $q\in\mathbb{Q}\cap A$ and every neighborhood $W$ of $(q,-q)$ there is $0<\delta\le\epsilon$ such that $q\times(-q,-q+\delta)\subset W\cap V$ .

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Section 31: Problem 9 Solution

Section 31: Problem 9 Solution