Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions

Terrence Anthony Edwards; James Edwards Joseph; Bhamini M. P. Nayar

doi:doi:10.11648/j.ajam.20210902.11

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Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions

Terrence Anthony Edwards, James Edwards Joseph, Bhamini M. P. Nayar

Published in American Journal of Applied Mathematics (Volume 9, Issue 2)

Received: 2 February 2021 Accepted: 23 February 2021 Published: 30 March 2021

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Abstract

E. Michael, in 1957, proved that the pracompactness is preserved by continuous closed functions from a space onto another. Michael’s proof is an immediate consequence of his characterization of paracompact spaces as those spaces with the property that each open cover of the space has a closure preserving refinement. Normality and transfinite induction were used to produce this characterization. J. M. Worrell, in 1985, proved, using the well-ordering principle, that continuous closed images of metacompact spaces are metacompact, as a consequence of a characterization of metacompact spaces he established earlier the same year. C. H. Dowker and R. N. Banerjee have provided the corresponding results for countable paracompactnes and countable metacompactness. In this article we extend these results for continuous, image closed and onto multifunctions. A result due to Joseph and Kwack that all open sets in Y have the form g(V) - g(X - V); where V is open in X, if g : X → Y is continuous, closed and onto (2006), is extended to image-closed, continuous, multifunctions. Such multifunctions as well as a characterization that a space is paracompact (metacompact) if and only if every ultrafilter of type P (M) converges, proved, in 1918, by Joseph and Nayar, is used to give generalizations of the invariance of paracompactness and metacompactness under continuous closed surjections to multifunctions.

Published in	American Journal of Applied Mathematics (Volume 9, Issue 2)
DOI	10.11648/j.ajam.20210902.11
Page(s)	38-43
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), 2021. Published by Science Publishing Group

Keywords

Multifunctions, Paracpmpact Spaces, Metacompact Spaces

References

[1]	R. N. Banerjee, Closed maps and countable metacompact spaces, J. London Math. Soc. 2 No. 8 (1974), 49-50.
[2]	A. Bella, A note on functionally compact spaces, Proc. Amer. Math. Soc. 98 No.1 (1986), 507 - 512.
[3]	S. W. Davis, Topology, McGraw Hill Companies, The Walter Rudin Student Series in Advanced Mathematics (2005)
[4]	R. Dickman and J. R. porter, _-closed subsets of Hausdorff spaces, Pacific Journal of Mathematics, 59 No.2 (1975), 407-415.
[5]	R. F. Dickman and A . Zame, functionally compact spaces, Pacific J. Math. 31 (1969), 303-311.
[6]	J. Dieudonn´e, Une g´en´eralisation des espaces compacts, J. Pures Appl. 23 (1944), 65-76.
[7]	C. H. Dowker, On countably paracompact spaces, Canad. J. Math. 3 No.2 (1951), 219 - 224.
[8]	R. Engelking, General Topology, PWN-Polish Scientific Publishers, Warszawa, Poland, (1977).
[9]	T.R. Hamlett, D. Rose and D. Jankovic Paracompactness with respect to an ideal, International Journal of Mathematics and Mathematical Sciences 20 No.3 (1997), 433-442.
[10]	J. E. Joseph, Multifunctions and graphs, Pac. J. Math. 79 (1978), 509-529.
[11]	J. E. Joseph and M. H. Kwack, A note on closed functions, Missouri J. Math. Sci. 18 (1), 59 -61, Winter 2006.
[12]	J. E. Joseph, M. H. Kwack and B. M. P. Nayar, Sequentially functionally compact and sequentially Ccompact spaces, Scientae Mathematicae 2 (2) (1999), 187-194.
[13]	J. E. Joseph and B. M. P. Nayar, New Proofs of Theorems of Michael and Worrell, Journal of Advanced Studies in Topology Vol 8:1 (2017), 21 - 23.
[14]	J. E. Joseph and B. M. P. Nayar, Ultrafilter chracerizations of paracompact, metacompact, paralindel´’of and meta- Lindel´’of spaces , International Journal of Pure and Applied Mathematics Vol.118, No.4 (2018) 1001-1005.
[15]	E. Michael, Another note on paracompactness, Proc. Amer. Math. Soc. 8 (1957) 822-828.
[16]	B. M. P. Nayar, A remark on C-compact spaces, J. Austral. Math. Soc.(Series A) 64 (1998), 327 - 328.
[17]	K. Sakai, Geometric Aspects of General Topology Springer Nature (2013).
[18]	T. B. Singh, Introduction to Topology, Springer Science and Business Media LLC (2019).
[19]	A. H. Stone, Paracompactness and product spaces, Bull. Amer. Math. Soc. 54 (1948), 977-992.
[20]	J.E. Vaughan, Linearly ordered collection of paracompactness, Proceedings of American Math. Soc.24 (1970), 186-192.
[21]	N. V. Veli˘cko, H-closed topological spaces, Math. Sb (N. S) 70 (112) (1966), 98 - 112(Russian), ( Amer. Math Soc. Transl. 78 (2) (1968), 103-118).
[22]	G. Viglino, C-compact spaces, Duke Math. J. 36(1969), 761-764.
[23]	J. M. Worrell, A characterization of metacompact spaces, Port. Math. 85 Fas. 3 (1985), 172-174.
[24]	J. M. Worrell, The closed continuous images of metacompact spaces, Port. Math., 85 Fas. 3 (1985), 175-179.

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APA Style

Terrence Anthony Edwards, James Edwards Joseph, Bhamini M. P. Nayar. (2021). Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions. American Journal of Applied Mathematics, 9(2), 38-43. https://doi.org/10.11648/j.ajam.20210902.11

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Terrence Anthony Edwards; James Edwards Joseph; Bhamini M. P. Nayar. Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions. Am. J. Appl. Math. 2021, 9(2), 38-43. doi: 10.11648/j.ajam.20210902.11

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AMA Style

Terrence Anthony Edwards, James Edwards Joseph, Bhamini M. P. Nayar. Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions. Am J Appl Math. 2021;9(2):38-43. doi: 10.11648/j.ajam.20210902.11

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@article{10.11648/j.ajam.20210902.11,
  author = {Terrence Anthony Edwards and James Edwards Joseph and Bhamini M. P. Nayar},
  title = {Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions},
  journal = {American Journal of Applied Mathematics},
  volume = {9},
  number = {2},
  pages = {38-43},
  doi = {10.11648/j.ajam.20210902.11},
  url = {https://doi.org/10.11648/j.ajam.20210902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20210902.11},
  abstract = {E. Michael, in 1957, proved that the pracompactness is preserved by continuous closed functions from a space onto another. Michael’s proof is an immediate consequence of his characterization of paracompact spaces as those spaces with the property that each open cover of the space has a closure preserving refinement. Normality and transfinite induction were used to produce this characterization. J. M. Worrell, in 1985, proved, using the well-ordering principle, that continuous closed images of metacompact spaces are metacompact, as a consequence of a characterization of metacompact spaces he established earlier the same year. C. H. Dowker and R. N. Banerjee have provided the corresponding results for countable paracompactnes and countable metacompactness. In this article we extend these results for continuous, image closed and onto multifunctions. A result due to Joseph and Kwack that all open sets in Y have the form g(V) - g(X - V); where V is open in X, if g : X → Y is continuous, closed and onto (2006), is extended to image-closed, continuous, multifunctions. Such multifunctions as well as a characterization that a space is paracompact (metacompact) if and only if every ultrafilter of type P (M) converges, proved, in 1918, by Joseph and Nayar, is used to give generalizations of the invariance of paracompactness and metacompactness under continuous closed surjections to multifunctions.},
 year = {2021}
}

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TY  - JOUR
T1  - Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions
AU  - Terrence Anthony Edwards
AU  - James Edwards Joseph
AU  - Bhamini M. P. Nayar
Y1  - 2021/03/30
PY  - 2021
N1  - https://doi.org/10.11648/j.ajam.20210902.11
DO  - 10.11648/j.ajam.20210902.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 38
EP  - 43
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20210902.11
AB  - E. Michael, in 1957, proved that the pracompactness is preserved by continuous closed functions from a space onto another. Michael’s proof is an immediate consequence of his characterization of paracompact spaces as those spaces with the property that each open cover of the space has a closure preserving refinement. Normality and transfinite induction were used to produce this characterization. J. M. Worrell, in 1985, proved, using the well-ordering principle, that continuous closed images of metacompact spaces are metacompact, as a consequence of a characterization of metacompact spaces he established earlier the same year. C. H. Dowker and R. N. Banerjee have provided the corresponding results for countable paracompactnes and countable metacompactness. In this article we extend these results for continuous, image closed and onto multifunctions. A result due to Joseph and Kwack that all open sets in Y have the form g(V) - g(X - V); where V is open in X, if g : X → Y is continuous, closed and onto (2006), is extended to image-closed, continuous, multifunctions. Such multifunctions as well as a characterization that a space is paracompact (metacompact) if and only if every ultrafilter of type P (M) converges, proved, in 1918, by Joseph and Nayar, is used to give generalizations of the invariance of paracompactness and metacompactness under continuous closed surjections to multifunctions.
VL  - 9
IS  - 2
ER  -

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Author Information

Terrence Anthony Edwards

Department of Mathematics University of the District of Columbia, Washington DC, USA
James Edwards Joseph

Department of Mathematics, Howard University, Washington DC, USA
Bhamini M. P. Nayar

Department of Mathematics Morgan State University, Baltimore, USA

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APA Style

Terrence Anthony Edwards, James Edwards Joseph, Bhamini M. P. Nayar. (2021). Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions. American Journal of Applied Mathematics, 9(2), 38-43. https://doi.org/10.11648/j.ajam.20210902.11

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ACS Style

Terrence Anthony Edwards; James Edwards Joseph; Bhamini M. P. Nayar. Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions. Am. J. Appl. Math. 2021, 9(2), 38-43. doi: 10.11648/j.ajam.20210902.11

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AMA Style

Terrence Anthony Edwards, James Edwards Joseph, Bhamini M. P. Nayar. Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions. Am J Appl Math. 2021;9(2):38-43. doi: 10.11648/j.ajam.20210902.11

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@article{10.11648/j.ajam.20210902.11,
  author = {Terrence Anthony Edwards and James Edwards Joseph and Bhamini M. P. Nayar},
  title = {Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions},
  journal = {American Journal of Applied Mathematics},
  volume = {9},
  number = {2},
  pages = {38-43},
  doi = {10.11648/j.ajam.20210902.11},
  url = {https://doi.org/10.11648/j.ajam.20210902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20210902.11},
  abstract = {E. Michael, in 1957, proved that the pracompactness is preserved by continuous closed functions from a space onto another. Michael’s proof is an immediate consequence of his characterization of paracompact spaces as those spaces with the property that each open cover of the space has a closure preserving refinement. Normality and transfinite induction were used to produce this characterization. J. M. Worrell, in 1985, proved, using the well-ordering principle, that continuous closed images of metacompact spaces are metacompact, as a consequence of a characterization of metacompact spaces he established earlier the same year. C. H. Dowker and R. N. Banerjee have provided the corresponding results for countable paracompactnes and countable metacompactness. In this article we extend these results for continuous, image closed and onto multifunctions. A result due to Joseph and Kwack that all open sets in Y have the form g(V) - g(X - V); where V is open in X, if g : X → Y is continuous, closed and onto (2006), is extended to image-closed, continuous, multifunctions. Such multifunctions as well as a characterization that a space is paracompact (metacompact) if and only if every ultrafilter of type P (M) converges, proved, in 1918, by Joseph and Nayar, is used to give generalizations of the invariance of paracompactness and metacompactness under continuous closed surjections to multifunctions.},
 year = {2021}
}

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TY  - JOUR
T1  - Michael, Dowker, Worrell, Banerjee Theorems Extended to Multifunctions
AU  - Terrence Anthony Edwards
AU  - James Edwards Joseph
AU  - Bhamini M. P. Nayar
Y1  - 2021/03/30
PY  - 2021
N1  - https://doi.org/10.11648/j.ajam.20210902.11
DO  - 10.11648/j.ajam.20210902.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 38
EP  - 43
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20210902.11
AB  - E. Michael, in 1957, proved that the pracompactness is preserved by continuous closed functions from a space onto another. Michael’s proof is an immediate consequence of his characterization of paracompact spaces as those spaces with the property that each open cover of the space has a closure preserving refinement. Normality and transfinite induction were used to produce this characterization. J. M. Worrell, in 1985, proved, using the well-ordering principle, that continuous closed images of metacompact spaces are metacompact, as a consequence of a characterization of metacompact spaces he established earlier the same year. C. H. Dowker and R. N. Banerjee have provided the corresponding results for countable paracompactnes and countable metacompactness. In this article we extend these results for continuous, image closed and onto multifunctions. A result due to Joseph and Kwack that all open sets in Y have the form g(V) - g(X - V); where V is open in X, if g : X → Y is continuous, closed and onto (2006), is extended to image-closed, continuous, multifunctions. Such multifunctions as well as a characterization that a space is paracompact (metacompact) if and only if every ultrafilter of type P (M) converges, proved, in 1918, by Joseph and Nayar, is used to give generalizations of the invariance of paracompactness and metacompactness under continuous closed surjections to multifunctions.
VL  - 9
IS  - 2
ER  -

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