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Hypographs of Upper Semi-continuous Maps and Continuous Maps on a Bounded Open Interval

Received: 23 March 2016     Published: 25 March 2016
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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)).

Published in American Journal of Applied Mathematics (Volume 4, Issue 2)
DOI 10.11648/j.ajam.20160402.12
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.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Hypograph, Upper Semi-continuous Maps, Continuous Maps, Bounded Open Interval, Hausdorff Metric, The Property of Strongly Universal

References
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[3] T. Dobrowolski, W. Marciszewski, J. Mogilski, On topopogical classification of function spaces Cp(X) of low Borel complexity, Trans. Amer. Math. Soc., vol. 678, pp. 307-324.
[4] Z. Q. Yang, “The hyperspace of the regions below of continuous maps is homeomorphic to c0,” Topology Appl., vol. 153, pp. 2908-2921.
[5] Z. Q. Yang, and X.E. Zhou, “A pair of spaces of upper semi-continuous maps and continuous maps,” Topology Appl. vol. 154, pp. 1737-1747.
[6] Z. Q. Yang and N. D. Wu, “The hyperspace of the regions below of continuous maps from S×S to I, Questions and Answers in General Topology,” vol. 26, pp. 29-39.
[7] Z. Q. Yang and N. D. Wu, “A topological position of the set of continuous maps in the set of upper semicontinuous maps,” Sci. China, Ser. A, vol. 52, pp. 1815-1828.
[8] Z. Yang, “The hyperspace of the regions below of all lattice-value continuous maps and its Hilbert cube compactification,” Sci. China Ser. A, vol. 48, pp. 469-484.
[9] Z. Q. Yang, and P. F. Yan, “Topological classification of function spaces with the Fell topology I,” Topology Appl., vol. 178, pp. 146–159.
[10] Z. Q. Yang, Y. M. Zheng, and J. Y. Chen, “Topological classification of function spaces with the Fell topology II,” Topology Appl., vol. 187, pp. 82–96.
[11] Z. Q. Yang, L. Z. Chen, and Y. M. Zheng, “Topological classification of function spaces with the Fell topology III,” Topology Appl., vol. 197, pp. 112–132.
[12] Y. J. Zhang and Z. Q. Yang, “Hyperspaces of the Regions Below of Upper Semi-continuous Maps on Non-compact Metric Spaces,” Advances in Mathematics (China), vol. 39, pp. 352-360.
[13] N. D. Wu, and Z. Q. Yang, “Spaces of Continuous Maps on a Class of Noncompact Metric Spaces,” Advances in Mathematics (China), vol 42, pp. 535-541.
[14] R. Cauty, and T. Dobrowolski, “Applying coordinate products to the topological identification of normed spaces,” Trans. Amer. Math. Soc., vol. 337, pp. 625-649.
[15] Van Mill J., The Infinite-Dimensional Topology of Function Spaces, Amsterdam: North-Holland Math. Library 64, Elsevier Sci. Publ. B. V., 2001.
[16] Van Mill J., Infinite-Dimensional Topology, Prerequisites and Introduction, Amsterdam: North-Holland Math. Library 43, Elsevier Sci. Publ. B. V., 1989.
[17] M. Bestvina, and J. Mogilski, “Characterizing certain incomplete infinite-dimensional absolute retracts,” Michigan Math. J., vol. 33, pp. 291-313.
[18] R. Cauty, T. Dobrowolski and W. Marciszewski, “A contribution to the topological classification of the spaces CP(X)”. Fund. Math. vol. 142, pp. 269-301.
[19] Z. Q. Yang, S. R. Hu, and G. Wei, “Topological structures of the space of continuous functions on a non-compact space with the Fell Topology,” Topology proceedings, vol. 41, pp. 17-38.
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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

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    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

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    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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  • @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}
    }
    

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    T1  - Hypographs of Upper Semi-continuous Maps and Continuous Maps on a Bounded Open Interval
    AU  - Nada Wu
    Y1  - 2016/03/25
    PY  - 2016
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    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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Author Information
  • Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, Guangdong, China

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