HEREDITARY SOBRIETY AND THE CARDINALITY OF T ₁ TOPOLOGIES ON COUNTABLE SETS

Abstract

A T ₀ space is called sober provided the only irreducibly closed sets are the closures of singletons; a closed set is irreducibly closed if it cannot be written as a union of two of its proper closed subsets. The relationship between hereditarily sober spaces and the lower separation axioms is examined; e.g., every hereditarily sober space satisfies axiom T _D (the derived set of every set is closed). For T ₁ spaces, hereditary sobriety is much weaker than Hausdorff, however an hereditarily sober T ₁ topology on a countably infinite set has cardinality of the continumn.