Section 17: Problem 18 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

Determine the closures of the following subsets of the ordered square: $\begin{eqnarray*} A & = & \{(1/n)\times0|n\in\mathbb{Z}_{+}\},\\ B & = & \{(1-1/n)\times\tfrac{1}{2}|n\in\mathbb{Z}_{+}\},\\ C & = & \{x\times0|0<x<1\},\\ D & = & \{x\times\tfrac{1}{2}|0<x<1\},\\ E & = & \{\tfrac{1}{2}\times y|0<y<1\}. \end{eqnarray*}$

The ordered square is $[0,1]\times[0,1]$ in the dictionary order topology. The closures are the unions of the sets and their limit points, so we describe limit points only. $A'=\{(0,1)\}$ as every neighborhood of $(0,1)$ is $((0,a),(b,c))$ for some $a<1$ and $b>0$ , which contains a point $(1/n,0)$ for some sufficiently large $n$ . Similarly, $B'=\{(1,0)\}$ , $C'=D'=(0,1]\times\{0\}\cup[0,1)\times\{1\}$ , and $E'=\{\tfrac{1}{2}\}\times[0,1]$ .

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Section 17: Problem 18 Solution

Section 17: Problem 18 Solution