Section 21: Problem 4 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

Show that $\mathbb{R}_{l}$ and the ordered square satisfy the first countability axiom. (This result does not, of course, imply that they are metrizable.)

For $x\in\mathbb{R}_{l}$ , an example of a countable basis at $x$ is $\{[x,x+\tfrac{1}{n})\}_{n\in\mathbb{Z}_{+}}$ .

For $a\times b\in I_{o}^{2}$ , an example of a countable basis at $a\times b$ is

if $b\in(0,1)$ , $\{((a,b-\tfrac{b}{n}),(a,b+\tfrac{1-b}{n}))\}_{n\in\mathbb{Z}_{+}}$ ,
if $b=0$ and $a>0$ , $\{((a-\tfrac{a}{n},1),(a,\tfrac{1}{n}))\}_{n\in\mathbb{Z}_{+}}$ ,
if $b=0$ and $a=0$ , $\{[(0,0),(0,\tfrac{1}{n}))\}_{n\in\mathbb{Z}_{+}}$ ,
if $b=1$ and $a<1$ , $\{((a,1-\tfrac{1}{n}),(a+\tfrac{1-a}{n},0))\}_{n\in\mathbb{Z}_{+}}$ ,
if $b=1$ and $a=1$ , $\{((1,1-\tfrac{1}{n}),(1,1)]\}_{n\in\mathbb{Z}_{+}}$ .

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Section 21: Problem 4 Solution

Section 21: Problem 4 Solution