AN ALTERNATE PROOF OF CLOSED MAPPING IN TOPOLOGY

  • Silas Ampem Kwame Nkrumah University of Science and Technology
Keywords: Closed mapping, Bijective mapping, Topological space, Continuous mapping, Open mapping

Abstract

The write-up seek to find alternative proof(s) to closed bijective mappings in topol-
ogy given that;

Given Definition
Let x and y be topological spaces and f : x → y a mapping f is called a closed
mapping if f(H) is closed in y for every closed set H in x.
Given Theorem
Let x and y be topological spaces and f : x → y a bijective mapping, the f is open ⇐⇒ f is closed.
In general function which are open need not to be closed and vice versa. In other
words, a mapping is closed if it carries closed sets over to closed sets.
This paper gives alternate proofs to closed bijective mappings in topology.

References

[1] Arkhangel’skii A. V. and Ponomarev V., in Fundamentals of General Topology: Problems and Exercises, Mathematics and Its Applications,p.xvi + 415, Hindustan Publishing Corporation, New Delhi, India, 1984. Last accessed, 10th October, 2019
[2] B. Margherita, ”Closed Map.” From MathWorld–A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/ClosedMap.html . Last accessed, 10th October, 2019
[3] Schochetman I. E., “A characterization of open mapping in terms of convergent sequences,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 76162, 5 pages, 2006. View at Google Scholar. Last accessed, 10th October, 2019
[4] Dugundji J. Topology, Prentice Hall, New Delhi, India, 1975. Last accessed, 10th October, 2019
[5] Lee J. M. (2003) Introduction to smooth Manifolds. Graduate Texts in Mathematics. 218. Springer Science and Business Media, pg. 550
[6] Munkres, J. R. (2000). Topology (2nd edition) Prentice hall, pg19
[7] John R. ”Continuous Map.” From MathWorld–A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/ContinuousMap.html.Last accessed, 10th October, 2019.
[8] Ponomarev V.I (originator),Encyclopedia of mathematics.
[9] https://www.emathzone.com/tutorials/general-topology/open-mapping-and-closed-mapping.html ixzz62vHwOXEd Last accessed, 20th November, 2019
Published
2019-12-30