Question 2(2.1) Let X = R and ( be a collection given by C = {0}U{Oc X : 0°

Answered: Let X = R and s be a collection given…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(2.1) Solution:

Let us consider $X = ℝ$ and $ζ = {\emptyset} \cup {O : O^{c} < \infty} = {\emptyset} \cup {O : O^{c} i s a f i n i t e s u b s e t o f ℝ}$ .

(a) We have to prove $(ℝ, ζ)$ forms a topological space.

<i> Distinctly $\emptyset, ℝ \in ζ$ . Since $ℝ^{c} = \emptyset$ is a finite set i.e. $ℝ^{c} < \infty$ , so $ℝ \in ζ$ .

<ii> Let ${U_{α}}_{α \in I}$ be an arbitrary collection of sets such that $U_{α} \in ζ \forall α \in I$ . Now $U_{α} \in ζ \forall α \in I \Rightarrow {(U_{α})}^{c} < \infty \forall α \in I$ . Then ${(\underset{α \in I}{\cup} U_{α})}^{c} = \underset{α \in I}{\cap} {(U_{α})}^{c} < \infty, \sin c e {(U_{α})}^{c} < \infty \forall α \in I$ . This proves that $\underset{α \in I}{\cup} U_{α} \in ζ$ .

<iii> Let I be a finite set such that $U_{α} \in ζ \forall α \in I$ . Now $U_{α} \in ζ \forall α \in I \Rightarrow {(U_{α})}^{c} < \infty \forall α \in I$ . Then ${(\underset{α \in I}{\cap} U_{α})}^{c} = \underset{α \in I}{\cup} {(U_{α})}^{c} < \infty, \sin c e {(U_{α})}^{c} < \infty \forall α \in I a n d I i s a f i n i t e s e t$ . This proves that $\underset{α \in I}{\cap} U_{α} \in ζ$ .