Hausdorff extensions

Abstract

Any space is considered to be a Hausdorff space. Let τ be an infinite cardinal. A point x ∈ X is called a P(τ )-point of the space X if for any non-empty family γ of open subsets of X for which x ∈ ∩γ and |γ| < τ there exists an open subset U of X such that x ∈ U ⊂ ∩γ. If any point of X is a P(τ )-point, then we say that P(τ )-space. Fix a set Φ of almost disjoint τ -centered families of subsets of the set E. We put eΦE = E ∪ Φ. On eΦE we construct two topologies.