Countable Compact Hausdorff Spaces Need Not Be Metrizable in Zf

We show that the existence of a countable, first countable, zerodimensional, compact Hausdorff space which is not second countable, hence not metrizable, is consistent with ZF. 1. Notation and terminology Definition 1.1. Let (X,T ) be a topological space and B a base for T . (i) X is said to be compact if every open cover U of X has a finite subcover V . X is said to be compact with respect to the base B if every open cover U ⊂ B of X has a finite subcover V . (ii) A collection U ⊂ ℘(X) is said to be locally finite (point finite) if every x ∈ X has an open neighborhood which meets only finitely many members of U (each x ∈ X belongs to only finitely many members of U). (iii) Let U be an open cover of X. A collection V ⊂ ℘(X) is said to be an open refinement of U if ⋃ V = X and each V ∈ V is open and is contained in some U ∈ U . (iv) X is said to be paracompact if every open cover U of X has a locally finite open refinement V . (v) X is said to be metacompact if every open cover U of X has a point finite open refinement V . (vi) X is said to be separable if it has a countable dense subset. (vii) X is said to be first countable if every x ∈ X has a countable open neighborhood base. (viii) X is said to be second countable if there is a countable base for T . (ix) X is said to be zero-dimensional if each of its points has a neighborhood base consisting of clopen (closed and open) sets. (x) X is said to be metrizable if there is a metric d on X such that the topology Td induced by d coincides with T . Received by the editors March 18, 2005 and, in revised form, May 15, 2005, November 5, 2005, and November 10, 2005. 2000 Mathematics Subject Classification. Primary 03E25, 03E35, 54A35, 54D10, 54D30, 54D35, 54D65, 54D70, 54E35.