### Abstract

An ideal on a set X is a nonempty collection of subsets of X which satisfies the following conditions (1)A ∈ I and B ⊂ A implies B ∈ I; (2) A ∈ I and B ∈ I implies A ∪ B ∈ I. Given a topological space (X,τ) an ideal I on X and A ⊂ X, R_{a}(A) is defined as ∪{U ∈ a: U - A ∈ I}, where the family of all a-open sets of X forms a topology [5, 6], denoted by τ^{a}. A topology, denoted τ^{a*}, finer than τ^{a} is generated by the basis β(I,τ) = {V - I: V ∈ τ^{a}(x), I ∈ I}, and a topology, denoted 〈R_{a}(τ)〉 coarser than τ^{a} is generated by the basis R_{a}(τ) = {R_{a}(U): U ∈ τ^{a}}. In this paper A bijection f: (X,τ,I) → (X,σ,J) is called a A*-homeomorphism if f: (X,τ^{a}*) → (Y,σ^{a}*) is a homeomorphism, R_{a}-homeomorphism if f: (X,R_{a}(τ)) → (Y,R_{a}(σ)) is a homeomorphism. Properties preserved by A*-homeomorphism are studied as well as necessary and sufficient conditions for a R_{a}-homeomorphism to be a A*-homeomorphism.

Original language | English |
---|---|

Pages (from-to) | 33-42 |

Number of pages | 10 |

Journal | Applied General Topology |

Volume | 15 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 |

### Fingerprint

### Keywords

- A*-homeomorphism
- a-local function
- Ideal spaces
- Topological groups

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Applied General Topology*,

*15*(1), 33-42. https://doi.org/10.4995/agt.2014.2126

**On topological groups via a-local functions.** / Al-Omeri, Wadei; Md. Noorani, Mohd. Salmi; Al-Omari, Ahmad.

Research output: Contribution to journal › Article

*Applied General Topology*, vol. 15, no. 1, pp. 33-42. https://doi.org/10.4995/agt.2014.2126

}

TY - JOUR

T1 - On topological groups via a-local functions

AU - Al-Omeri, Wadei

AU - Md. Noorani, Mohd. Salmi

AU - Al-Omari, Ahmad

PY - 2014

Y1 - 2014

N2 - An ideal on a set X is a nonempty collection of subsets of X which satisfies the following conditions (1)A ∈ I and B ⊂ A implies B ∈ I; (2) A ∈ I and B ∈ I implies A ∪ B ∈ I. Given a topological space (X,τ) an ideal I on X and A ⊂ X, Ra(A) is defined as ∪{U ∈ a: U - A ∈ I}, where the family of all a-open sets of X forms a topology [5, 6], denoted by τa. A topology, denoted τa*, finer than τa is generated by the basis β(I,τ) = {V - I: V ∈ τa(x), I ∈ I}, and a topology, denoted 〈Ra(τ)〉 coarser than τa is generated by the basis Ra(τ) = {Ra(U): U ∈ τa}. In this paper A bijection f: (X,τ,I) → (X,σ,J) is called a A*-homeomorphism if f: (X,τa*) → (Y,σa*) is a homeomorphism, Ra-homeomorphism if f: (X,Ra(τ)) → (Y,Ra(σ)) is a homeomorphism. Properties preserved by A*-homeomorphism are studied as well as necessary and sufficient conditions for a Ra-homeomorphism to be a A*-homeomorphism.

AB - An ideal on a set X is a nonempty collection of subsets of X which satisfies the following conditions (1)A ∈ I and B ⊂ A implies B ∈ I; (2) A ∈ I and B ∈ I implies A ∪ B ∈ I. Given a topological space (X,τ) an ideal I on X and A ⊂ X, Ra(A) is defined as ∪{U ∈ a: U - A ∈ I}, where the family of all a-open sets of X forms a topology [5, 6], denoted by τa. A topology, denoted τa*, finer than τa is generated by the basis β(I,τ) = {V - I: V ∈ τa(x), I ∈ I}, and a topology, denoted 〈Ra(τ)〉 coarser than τa is generated by the basis Ra(τ) = {Ra(U): U ∈ τa}. In this paper A bijection f: (X,τ,I) → (X,σ,J) is called a A*-homeomorphism if f: (X,τa*) → (Y,σa*) is a homeomorphism, Ra-homeomorphism if f: (X,Ra(τ)) → (Y,Ra(σ)) is a homeomorphism. Properties preserved by A*-homeomorphism are studied as well as necessary and sufficient conditions for a Ra-homeomorphism to be a A*-homeomorphism.

KW - A-homeomorphism

KW - a-local function

KW - Ideal spaces

KW - Topological groups

UR - http://www.scopus.com/inward/record.url?scp=84901940350&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84901940350&partnerID=8YFLogxK

U2 - 10.4995/agt.2014.2126

DO - 10.4995/agt.2014.2126

M3 - Article

VL - 15

SP - 33

EP - 42

JO - Applied General Topology

JF - Applied General Topology

SN - 1576-9402

IS - 1

ER -