A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R is equipped with the usual topology, then the boundedness and the closedness are not topological properties because * Ris homeomorphic to ]-co, O[ Ris homeomorphic to Ja,b[ O [a,b] is not homeomorphic to Ja,b[ Ris homeomorphic to ]-o, 0]

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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neither closed in (Y,Ty) nor in R
closed in (Y,Ty) and not closed in R
not closed in (Y,Ty) and closed in R
closed in (Y,Ty) and closed in R
A property is said to be a topological
property if it is preserved by
homeomorphism. Suppose that R is
equipped with the usual topology, then the
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 ]-, 0]
Let X be a discrete spaces then *
지
O X is never homeomorphic to R
Transcribed Image Text:neither closed in (Y,Ty) nor in R closed in (Y,Ty) and not closed in R not closed in (Y,Ty) and closed in R closed in (Y,Ty) and closed in R A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R is equipped with the usual topology, then the 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 ]-, 0] Let X be a discrete spaces then * 지 O X is never homeomorphic to R
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