Section 17: Problem 16 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

Consider the five topologies on $\mathbb{R}$ given in Exercise 7 of § 13.

(a) Determine the closure of the set $К=\{1/n|n\in\mathbb{Z}_{+}\}$ under each of these topologies.

(b) Which of these topologies satisfy the Hausdorff axiom? the $T_{1}$ axiom?

(a) and (b) The closure is the union of the set and its limit points. So we describe only limit points.

Topology:	$\mathcal{T}_{1}=\mathbb{R}$	$\mathcal{T}_{2}=\mathbb{R}_{K}$	$\mathcal{T}_{3}=\mathbb{R}_{fc}$	$\mathcal{T}_{4}=\mathbb{R}_{u}$	$\mathcal{T}_{5}=\mathbb{R}_{-\infty}$
$K'$	$\{0\}$	$\emptyset$	$\mathbb{R}$	$\emptyset$	$\overline{\mathbb{R}}_{+}$
Hausdorff	+	+	-	+	-
$T_{1}$	+	+	+	+	-

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Section 17: Problem 16 Solution

Section 17: Problem 16 Solution