[Paper Review] Idealization of Ganster-Reilly decomposition theorems

Julian Dontchev|ArXiv.org|Jan 5, 1999

Fuzzy and Soft Set Theory12 references37 citations

TL;DR

This paper extends Ganster–Reilly's decomposition of continuity by introducing ideal-based topological structures, proving that a function between ideal topological spaces is continuous if and only if it is both pre-ℐ-continuous and ℐ-locally closed (ℐ-LC)-continuous. The key contribution is an idealized version of the classical Ganster–Reilly theorem, generalizing continuity decompositions via ℐ-open sets and the local function A* with respect to a topological ideal ℐ.

ABSTRACT

In 1990, Ganster and Reilly proved that a function is continuous if and only if it is precontinuous and LC-continuous. In this paper we extend their decomposition of continuity in terms of ideals. We show that a function $f \colon (X,τ,{\cal I}) o (Y,σ)$ is continuous if and only if it is pre-I-continuous and I-LC-continuous. We also provide a decomposition of I-continuity.

Motivation & Objective

• To generalize the classical Ganster–Reilly decomposition of continuity to ideal topological spaces.
• To define and analyze ℐ-continuity, pre-ℐ-continuity, and ℐ-LC-continuity in the context of topological ideals.
• To establish a new decomposition theorem for continuity using ideals, extending the original result from 1990.
• To investigate the relationship between various continuity types under ideal topologies, particularly in Hayashi-Samuels spaces.
• To provide a unified framework for continuity decompositions via the local function A* and ℐ-open sets.

Proposed method

• Introduces the concept of an ideal topological space (X, τ, ℐ), where ℐ is a proper ideal on X satisfying heredity and finite additivity.
• Defines the local function A* = {x ∈ X : for every U ∈ τ(x), U ∩ A ∉ ℐ}, which generalizes closure, ω-accumulation, and condensation points.
• Introduces ℐ-open sets via A ⊆ Int(A*), and defines ℐ-continuous, pre-ℐ-continuous, and ℐ-LC-continuous functions.
• Uses the topology τ* generated by {U ∖ I : U ∈ τ, I ∈ ℐ} and the Kuratowski closure operator Cl*(A) = A ∪ A*.
• Applies the Hayashi-Samuels condition (X = X*) to ensure consistency between τ and τ*, enabling the decomposition.
• Proves that in Hayashi-Samuels spaces, continuity is equivalent to the conjunction of pre-ℐ-continuity and ℐ-LC-continuity.

Experimental results

Research questions

• RQ1 Can the classical Ganster–Reilly decomposition of continuity be generalized to ideal topological spaces?
• RQ2 How do ℐ-continuity, pre-ℐ-continuity, and ℐ-LC-continuity relate in ideal topological spaces?
• RQ3 Under what conditions does ℐ-LC-continuity imply continuity when combined with pre-ℐ-continuity?
• RQ4 What is the role of the local function A* in defining continuity decompositions under ideals?
• RQ5 How do standard ideals (e.g., ℐ = {∅}, ℐ = ℳ, ℐ = 𝒩) recover known continuity decompositions?

Key findings

• A function f: (X, τ, ℐ) → (Y, σ) is continuous if and only if it is both pre-ℐ-continuous and ℐ-LC-continuous in a Hayashi-Samuels space.
• The idealized decomposition generalizes the original Ganster–Reilly theorem, recovering it as special cases when ℐ = {∅} or ℐ = 𝒩.
• The class of ℐ-locally closed sets is defined as A = U ∩ V with U open and V ⋆-perfect, generalizing locally closed sets.
• The local function A* generalizes closure (ℐ = {∅}), ω-accumulation (ℐ = ℱ), and condensation points (ℐ = ℂ).
• In the case ℐ = ℳ (meager sets), ℐ-LC-continuity corresponds to functions whose preimages are inexhaustibly approached.
• The paper constructs counterexamples showing that pre-ℐ-continuity and ℐ-LC-continuity are independent of ℐ-continuity, and neither implies the other.