Stackbases in power sets of neighbourhood spaces preserving the continuity of mappings
Jan Chvalina · Czech digital mathematics library · 1981
A generalized filter base in a set M is defined in [7] as a nonempty family of nonempty subsets of M. The generalized filter base is called a proper family in [4] and a stackbase in [3].We shall use the last term.If (M, c x ) 9 (M, c 2 ) are topological spaces determined by Kuratowski closure operations c t , c 2 then there exist stackbases a t , (M, c 2 ) whenever Xea 2 then/" l (X)e tr, or/" X (X) = 0. Further, for each xeM and each X 2 ea 2 r\ [/(*)) (where \x) = {X c M : x e X}) there exists a set X t e ea t n [*) with/(X t ) c X 2 .Moreover, the assignment c -> a is one-to-one.Indeed, assigning to a Kuratowski closure operation c on M the system a of all nonempty open subsets of the topological space (M, c) we obtain that a n [*) is a neighbourhood base at the point x and the above statements follow e.g. from [6] Theorem 1.4.6.The just formulated continuity condition at JC e M can be written in the form a 2 n n [f(x)) <f(a x n [*)), where A, -< X 2 for X u X 2 cz exp M means, by [4], that for each Xe X 1 there exists YeX 2 with Y cz X This note aims to show that the above described assertion does not hold in the case of neighbourhood spaces ([5], [7]) which are not topological, i.e. corresponding closure operations are the so called Fr6chet-Cech closure operations ([1], [5]satisfying the following three axioms only: 1° c0 = 0, 2° X c cX, 3° X cr Yimplies cX c cY).Further we shall prove the existence of an assignment of a stackbase $f(t) in exp M to an arbitrary Frechet-Cech closure operation t on M with the following properties: If t t ^ t 2 then Sf(t x ) # £f(t 2 ) and for every continuous mapping/of the neighbourhood space (M, t x ) into the neighbourhood space (M, t 2 ) the corresponding self-map/of exp M satisfies the condition/ _1 (X) e Sf(t x ) u {0} for each Xe Sf(f%\ consequently $f(t 2 ) n [/(X))