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Corrigendum to “On Irresolute-Topological Vector Spaces”

Madhu Ram , Amjad Hussain
Turkish Journal of Analysis and Number Theory. 2018, 6(5), 148-148. DOI: 10.12691/tjant-6-5-4
Received July 19, 2018; Revised August 24, 2018; Accepted October 14, 2018

Abstract

In this note, we disprove two results in reference [1].

Theorem 5 and part (a) of Theorem 6 in the published paper 1 are not correct. Indeed, in the proof of Theorem 5 in 1, it is not necessary that the set V1 contains any interval of the form (−ϵ, ϵ). To assert this, it suffices to give a semi-open set V1 in ℝ which does not contain (−ϵ, ϵ), for any ϵ > 0. E.g. consider the set V1 = [0, 1). Then V1 ∈ N0(ℝ) which does not include any set of the form (−ϵ, ϵ).

Alternatively, consider the example [ 1, Example 1] of irresolute topological vector spaces. Let U = [0, 1) ⊆ X = ℝ. Then U ∈ N0(X) but U is not absorbing.

Next, on review of [ 1, Theorem 6], it is figured out that part (a) of Theorem 6 in 1 is incorrect. The flaws are observed from the proof of this theorem. Here the author believes that the set V1 ∈ N0(ℝ) contains an interval of the form (−ϵ, ϵ) which is not always true as we have seen above.

Acknowledgements

The authors are grateful to the referee and the reviewers for their valuable comments and suggestions.

References

[1]  T. Al-Hawary and A. Al-Nayef, On Irresolute-Topological Vector Spaces, Math. Sci. Res. Hot-Line, 5 (2001), 49-53.
In article      
 

Published with license by Science and Education Publishing, Copyright © 2018 Madhu Ram and Amjad Hussain

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

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Normal Style
Madhu Ram, Amjad Hussain. Corrigendum to “On Irresolute-Topological Vector Spaces”. Turkish Journal of Analysis and Number Theory. Vol. 6, No. 5, 2018, pp 148-148. http://pubs.sciepub.com/tjant/6/5/4
MLA Style
Ram, Madhu, and Amjad Hussain. "Corrigendum to “On Irresolute-Topological Vector Spaces”." Turkish Journal of Analysis and Number Theory 6.5 (2018): 148-148.
APA Style
Ram, M. , & Hussain, A. (2018). Corrigendum to “On Irresolute-Topological Vector Spaces”. Turkish Journal of Analysis and Number Theory, 6(5), 148-148.
Chicago Style
Ram, Madhu, and Amjad Hussain. "Corrigendum to “On Irresolute-Topological Vector Spaces”." Turkish Journal of Analysis and Number Theory 6, no. 5 (2018): 148-148.
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[1]  T. Al-Hawary and A. Al-Nayef, On Irresolute-Topological Vector Spaces, Math. Sci. Res. Hot-Line, 5 (2001), 49-53.
In article