Nearness structures and proximity extensions

Abstract

Proximity, contiguity and nearness structures are here studied from a unified point of view. In the discussion the role that grills can play in the theory is emphasized. Nearness structures were recently introduced by Herrlich and Naimpally. Thron pointed out the importance of grills in proximity theory. Nearness structures <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v"> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding="application/x-tex">v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are then used to generate proximity extensions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis phi comma left-parenthesis upper X Superscript v Baseline comma normal upper Pi Superscript v Baseline right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ϕ </mml:mi> <mml:mo>,</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>v</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi mathvariant="normal"> Π </mml:mi> <mml:mi>v</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\phi ,({X^v},{\Pi ^v}))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a given LO-proximity space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma normal upper Pi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> Π </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,\Pi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi Subscript v Baseline equals normal upper Pi"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> Π </mml:mi> <mml:mi>v</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal"> Π </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Pi _v} = \Pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Finally, the relation of the extensions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis phi comma left-parenthesis upper X Superscript v Baseline comma normal upper Pi Superscript v Baseline right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ϕ </mml:mi> <mml:mo>,</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>v</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi mathvariant="normal"> Π </mml:mi> <mml:mi>v</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\phi ,({X^v},{\Pi ^v}))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to arbitrary extensions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis i comma left-parenthesis upper Y comma normal upper Pi Superscript asterisk Baseline right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi mathvariant="normal"> Π </mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(i,(Y,{\Pi ^ \ast }))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is investigated.