For any bounded open interval X in the Euclidean space E1, let ↓USC(X) and ↓C(X) be the families of all hypographs of upper semi-continuous maps and continuous maps from X to I=[0,1], respectively. They are endowed with the topology induced by the Hausdorff metric of the metric space Y×I，Y is the closure of X. It was proved in other two papers respectively that ↓USC(X) and ↓C(X) are homeomorphic to s and c0 respectively, where s=(-1,1)∞ and c0={(xn)ϵ(-1,1) ∞: limn→ ∞ xn=0}. However the topological structure of the pair (↓USC(X), ↓C(X)) was not clear. In the present paper, it is proved in the strongly universal method that the pair of spaces (↓USC(X), ↓C(X)) is pair homeomorphic to (s∞,c0∞) which is not homeomorphic to (s, c0). Hence this paper figures out the topological structure of the pair (↓USC(X), ↓C(X)).
| DOI | 10.11648/j.ajam.20160402.12 |
| Published in | American Journal of Applied Mathematics (Volume 4, Issue 2, April 2016) |
| Page(s) | 75-79 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Hypograph, Upper Semi-continuous Maps, Continuous Maps, Bounded Open Interval, Hausdorff Metric, The Property of Strongly Universal
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APA Style
Nada Wu. (2016). Hypographs of Upper Semi-continuous Maps and Continuous Maps on a Bounded Open Interval. American Journal of Applied Mathematics, 4(2), 75-79. https://doi.org/10.11648/j.ajam.20160402.12
ACS Style
Nada Wu. Hypographs of Upper Semi-continuous Maps and Continuous Maps on a Bounded Open Interval. Am. J. Appl. Math. 2016, 4(2), 75-79. doi: 10.11648/j.ajam.20160402.12
AMA Style
Nada Wu. Hypographs of Upper Semi-continuous Maps and Continuous Maps on a Bounded Open Interval. Am J Appl Math. 2016;4(2):75-79. doi: 10.11648/j.ajam.20160402.12
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Download@article{10.11648/j.ajam.20160402.12,
author = {Nada Wu},
title = {Hypographs of Upper Semi-continuous Maps and Continuous Maps on a Bounded Open Interval},
journal = {American Journal of Applied Mathematics},
volume = {4},
number = {2},
pages = {75-79},
doi = {10.11648/j.ajam.20160402.12},
url = {https://doi.org/10.11648/j.ajam.20160402.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20160402.12},
abstract = {For any bounded open interval X in the Euclidean space E1, let ↓USC(X) and ↓C(X) be the families of all hypographs of upper semi-continuous maps and continuous maps from X to I=[0,1], respectively. They are endowed with the topology induced by the Hausdorff metric of the metric space Y×I，Y is the closure of X. It was proved in other two papers respectively that ↓USC(X) and ↓C(X) are homeomorphic to s and c0 respectively, where s=(-1,1)∞ and c0={(xn)ϵ(-1,1) ∞: limn→ ∞ xn=0}. However the topological structure of the pair (↓USC(X), ↓C(X)) was not clear. In the present paper, it is proved in the strongly universal method that the pair of spaces (↓USC(X), ↓C(X)) is pair homeomorphic to (s∞,c0∞) which is not homeomorphic to (s, c0). Hence this paper figures out the topological structure of the pair (↓USC(X), ↓C(X)).},
year = {2016}
}
TY - JOUR
T1 - Hypographs of Upper Semi-continuous Maps and Continuous Maps on a Bounded Open Interval
AU - Nada Wu
Y1 - 2016/03/25
PY - 2016
N1 - https://doi.org/10.11648/j.ajam.20160402.12
DO - 10.11648/j.ajam.20160402.12
T2 - American Journal of Applied Mathematics
JF - American Journal of Applied Mathematics
JO - American Journal of Applied Mathematics
SP - 75
EP - 79
PB - Science Publishing Group
SN - 2330-006X
UR - https://doi.org/10.11648/j.ajam.20160402.12
AB - For any bounded open interval X in the Euclidean space E1, let ↓USC(X) and ↓C(X) be the families of all hypographs of upper semi-continuous maps and continuous maps from X to I=[0,1], respectively. They are endowed with the topology induced by the Hausdorff metric of the metric space Y×I，Y is the closure of X. It was proved in other two papers respectively that ↓USC(X) and ↓C(X) are homeomorphic to s and c0 respectively, where s=(-1,1)∞ and c0={(xn)ϵ(-1,1) ∞: limn→ ∞ xn=0}. However the topological structure of the pair (↓USC(X), ↓C(X)) was not clear. In the present paper, it is proved in the strongly universal method that the pair of spaces (↓USC(X), ↓C(X)) is pair homeomorphic to (s∞,c0∞) which is not homeomorphic to (s, c0). Hence this paper figures out the topological structure of the pair (↓USC(X), ↓C(X)).
VL - 4
IS - 2
ER -
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