Acta Math. Hungar. DOI: 0 A NOTE ON SPACES WITH A COUNTABLE μ-BASE M. ARAR Salman Bin Abdulaziz University, Math. Depart. at Al-Aflaj, Saudi Arabia e-mail: muradshhada@gmail.com (Received January 30, 2014; revised April 14, 2014; accepted April 15, 2014) Abstract. In topological spaces every normal space with a countable base is metacompact. We show that this is not necessarily true in generalized topological spaces; more exactly we give an example of a μ-normal space with a countable μ-base which has a μ-open cover with no μ-open point-finite refinement. 1. Preliminaries Definitions and theorems mentioned in this section can be found in [1], [2], [3], [4] and [5], for more details one can consult them. Let X be a nonempty set. A collection μ of subsets of X is called a gen- eralized topology on X and the pair (X,μ) is called a generalized topological space, if μ satisfies the following two conditions: (1) ∅∈ μ. (2) Any union of elements of μ belongs to μ. Let β ⊂ exp(X ) and ∅∈ β . Then β is called a base for μ if μ = {  β ′ ; β ′ ⊂ β } . We also say μ is generated by β . A generalized topological space (X, μ) is said to be strong if X ∈ μ. A subset B of X is called μ-open (resp. μ-closed) if B ∈ μ (resp. if X - B ∈ μ). The set of all μ-open sets containing a point x ∈ X will be denoted by μ x (i.e. μ x = {U ∈ μ; x ∈ U }). Definition 1.1. Let X be a strong generalized topological space. Then (1) X is called a μT 1 -space if and only if for every x, y ∈ X with x = y there exist U x ∈ μ x and U y ∈ μ y such that y ∈ U x and x ∈ U y . (2) X is called a μT 2 -space if and only if for every x, y ∈ X with x = y there exist U x ∈ μ x and U y ∈ μ y such that U x  U y = ∅. (3) X is called a μT 3 -space if and only if for every x ∈ X and every μ- closed subset F of X with x ∈ F there exist U x ∈ μ x and U F ∈ μ such that F ⊂ U F and U x  U F = ∅. Key words and phrases: generalized topological space, μ-compact, μ-metacompact, countable μ-base, μ-paracompact, μ-separation, μ-Lindel¨of, μ-locally finite, μ-point finite, μ-open cover. Mathematics Subject Classification: primary 54A05, 54D10, 54D15, 54D30. 0236-5294/$20.00 © 2014 Akade ´miai Kiado ´, Budapest, Hungary Acta Math. Hungar. DOI: 10.1007/s10474-014-0434-0