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