Lebesgue sets and insertion of a continuous function

Lane, Ernest P.

doi:10.1090/s0002-9939-1983-0684654-0

Cited by 6 publications

(2 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Both the KatEtov-Tong and Tietze-Urysohn theorems hold true for normal L-fuzzy topological spaces, as proved in [17] by utilizing Kat6tov's method of proof (see also [18] for further generalization a la BLAIR [3] and LANE [21]). We notice that the techniques of BONAN and SCOTT work in the fuzzy situation as well.…”

Section: (C)])mentioning

confidence: 99%

“…It is shown in [15] that, in fact, the abstract versions of those characterizations are inherited by O-rings of subsets and measurable real-valued functions. These versions include the insertion theorem of BLAIR [3] and LANE [21], the extension theorem of Mrbwka [22], and the Urysohn Extension Theorem of GILLMAN and JERISON [8]. The proofs of [15] can be applied, mutatis mutandis, to O-rings in L x and measurable @L)-valued functions.…”

Section: +Rings Of L-fuzzy Sets and Measurable Functionsmentioning

confidence: 99%

See 1 more Smart Citation

Inserting L‐fuzzy Real‐valued Functions

Kotzé

Kubiak

1993

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

Section: (C)])mentioning

confidence: 99%

Section: +Rings Of L-fuzzy Sets and Measurable Functionsmentioning

confidence: 99%

Inserting L‐fuzzy Real‐valued Functions

Kotzé

Kubiak

1993

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

A separation theorem for semicontinuous functions on preordered topological spaces

Edwards¹

2005

Arch. Math.

View full text Add to dashboard Cite

Suppose that X is a topological space with preorder , and that −g, f are bounded upper semicontinuous functions on X such that g(x) f (y) whenever x y. We consider the question whether there exists a bounded increasing continuous function h on X such that g h f , and obtain an existence theorem that gives necessary and sufficient conditions. This result leads to an extension theorem giving conditions that allow a bounded increasing continuous function defined on an open subset of X to be extended to a function of the same type on X. The application of these results to extremally disconnected locally compact spaces is studied. Introduction.Before explaining the purpose of this paper it will be convenient to make a number of definitions. By a preorder for a set X we shall mean a reflexive, transitive relation in X, and by an order for X we shall mean an antisymmetric preorder. We shall use the symbol to denote a preorder in X, but also to denote the usual order relation for real numbers or real-valued functions. Suppose we are given a preorder for X. We say that a real-valued function f on X is increasing if f (x) f (y) whenever x, y ∈ X and x y. A subset S of X is said to be increasing if its indicator function 1 S is increasing. Decreasing functions and sets are defined analogously. Note that the complement of an increasing set is decreasing and vice versa. Given a bounded real function f on X, we denote by f ↑ the least increasing real function on X that majorizes f (see §2). Given a subset T of X we denote by i(T ) the smallest increasing subset of X that contains T and by d(T ) the smallest decreasing subset of X that contains T . If X is also a topological space, we denote by C b (X, ) the set of bounded increasing real-valued continuous functions on X, and for a bounded real function f on X we denote by f and f respectively the lower and the upper semicontinuous regularizations of f (see, for example, [4]). Finally, by a clopen set in a topological space we shall mean any subset that is open and closed, and we shall denote by E(X, ) the set of all increasing clopen subsets of X.

show abstract

A new look at some classical theorems on continuous functions on normal spaces

García

Kubiak

2008

Acta Math Hung

View full text Add to dashboard Cite

A sucient condition for the strict insertion of a continuous function between two comparable upper and lower semicontinuous functions on a normal space is given. Among immediate corollaries are the classical insertion theorems of Michael and Dowker. Our insertion lemma also provides purely topological proofs of some standard results on closed subsets of normal spaces which normally depend upon uniform convergence of series of continuous functions. We also establish a Tietze-type extension theorem characterizing closed G δ -sets in a normal space.

show abstract

Lebesgue sets and insertion of a continuous function

Cited by 6 publications

References 13 publications

Inserting L‐fuzzy Real‐valued Functions

Inserting L‐fuzzy Real‐valued Functions

A separation theorem for semicontinuous functions on preordered topological spaces

A new look at some classical theorems on continuous functions on normal spaces

Contact Info

Product

Resources

About