boundedness and the closedness are nottopological properties because *Ris homeomorphic to ]-o, O[Ris homeomorphic to ]a,b[[a,b] is not homeomorphic to ]a,b[Ris homeomorphic to ]-o, 0]Let X be a discrete spaces thenX is never homeomorphic to RX is homeomorphic to R if and only if X iscountableX is homeomorphic to R if and only if X isinfiniteX is homeomorphic to R if and only if X isfiniteWe define the included point topology byTp={UCR;U=Ø or pEU}. Let A = [3,5[, thenA is dense in R if *Ris equipped with the usual topology

A property of a topological space is called a homeomorphic property if it remains invariant under…

Answered: poundedness and the closedness are not…

poundedness and the closedness are not opological properties because * O Ris homeomorphic to ]-00, O[ O Ris homeomorphic to Ja,b[ O [a,b] is not homeomorphic to Ja,b[ ORis homeomorphic to ]-00, 0]

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.1: Postulates For The Integers (optional)

Problem 11E

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Question

boundedness and the closedness are not
topological properties because *
Ris homeomorphic to ]-o, O[
Ris homeomorphic to ]a,b[
[a,b] is not homeomorphic to ]a,b[
Ris homeomorphic to ]-o, 0]
Let X be a discrete spaces then
X is never homeomorphic to R
X is homeomorphic to R if and only if X is
countable
X is homeomorphic to R if and only if X is
infinite
X is homeomorphic to R if and only if X is
finite
We define the included point topology by
Tp={UCR;U=Ø or pEU}. Let A = [3,5[, then
A is dense in R if *
Ris equipped with the usual topology

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