6533b7ddfe1ef96bd1274b42

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Erratum to “Irregularity” [Topology Appl. 154 (8) (2007) 1565–1580]

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[2, Proposition 4.4] states that each regular pretopology is topologically regular. Professor F. Mynard (Georgia Southern University) advised the authors that he was not convinced by the proof of that proposition, which enabled us to realize the proposition is wrong, as the example below shows. Recall that (e.g., [2]) a pretopology ξ on a set X is called regular if Vξ (x)⊂ adh ξ Vξ (x) (respectively, topologically regular if Vξ (x)⊂ cl ξ Vξ (x)) for every x ∈ X . As a consequence, in the sequel of [2], regular should be read topologically regular in a few instances, in particular in [2, Theorem 4.6]. [2, Proposition 4.4] is also quoted in [3], where it is used in some reformulations of classical facts on weakly regular topologies like Theorem 3.1. In terms of pretopologies, a topology τ is weakly regular if there exists a topologically regular pretopology ξ such that τ is the topological reflection T ξ of ξ. Accordingly, in this context, topologically was sometimes omitted. Except for these reformulations, the discussed proposition is not used in [3].