𝐻-closed extensions. II (1975)

The internal structure and external properties (in terms of other <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-closed extensions) of the Fomin extension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma upper X"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a Hausdorff space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are investigated. The relationship between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma upper X"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Stone-Čech compactification of the absolute of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is developed and used to prove that a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma upper X"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-closed subset of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma upper X minus upper X"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>X</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma X\backslash X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact and to show the existence of a Tychonoff space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma upper X minus upper X"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>X</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma X\backslash X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is homeomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta upper Y minus upper Y"> <mml:semantics> <mml:mrow> <mml:mi>β<!-- β --></mml:mi> <mml:mi>Y</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta Y\backslash Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The sequential closure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma upper X"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is shown to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is known that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma upper X"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not necessarily projectively larger than any other strict <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-closed extension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; a necessary and sufficient condition is developed to determine when a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-closed extension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is projectively smaller then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma upper X"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A theorem by Magill is extended by showing that the sets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta"> <mml:semantics> <mml:mi>θ<!-- θ --></mml:mi> <mml:annotation encoding="application/x-tex">\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-isomorphism classes of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-closed extensions of locally <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-closed spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are lattice isomorphic if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma upper X minus upper X"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>X</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma X\backslash X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma upper Z minus upper Z"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>Z</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖<!-- ∖ --></mml:mi> <mml:mi>Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma Z\backslash Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are homeomorphic. Harris has characterized those simple Hausdorff extensions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are subextensions of the Katětov extension. Characterizations