Lattices of topological extensions

Abstract

For completely regular Hausdorff spaces, we consider topological properties <italic>P</italic> which are akin to compactness in the sense of Herrlich and van der Slot and satisfy the equivalent of Mrowka’s condition (W). The algebraic structure of the family of tight extensions of <italic>X</italic> (which have <italic>P</italic> and contain no proper extension with that property) is studied. Where <italic>X</italic> has <italic>P</italic> locally but not globally, the relations between the complete lattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Superscript asterisk Baseline left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>P</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{P^ \ast }(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of those tight extensions which are above the maximal one-point extension and the topology of the <italic>P</italic> -reflection are investigated and conditions found under which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Superscript asterisk Baseline left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>P</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{P^\ast }(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> characterizes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma upper X minus upper X"> <mml:semantics> <mml:mrow> <mml:mi> γ </mml:mi> <mml:mi>X</mml:mi> <mml:mo> − </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\gamma X - X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The results include those of Magill on the lattice of compactifications of a locally compact space, and other applications are considered.