Karim Belaid - Academia.edu (original) (raw)

Papers by Karim Belaid

Research paper thumbnail of SUBMAXIMAL AND SPECTRAL SPACES

Mathematical Proceedings of The Royal Irish Academy, 2008

A topological space X is said to be submaximal if every dense subset of X is open. In this paper,... more A topological space X is said to be submaximal if every dense subset of X is open. In this paper, descriptions of submaximal spectral spaces and Stone submaximal spaces are given. Throughout this paper a number of illustrative examples are given.

Research paper thumbnail of On a conjecture about spectral sets

Topology and Its Applications, 2004

Let R be a commutative ring with identity. We denote by Spec(R) the set of prime ideals of R. Cal... more Let R be a commutative ring with identity. We denote by Spec(R) the set of prime ideals of R. Call a partial ordered set spectral if it is order isomorphic to (Spec(R), ⊆) for some R. A longstanding open question about spectral sets (since 1976), is that of Lewis and Ohm [Canad. J. Math. 28 (1976) 820, Question 3.4]: "If (X, ) is an ordered disjoint union of the posets (X λ , λ ), λ ∈ Λ, and if (X, ) is spectral, then are the (X λ , λ ) also spectral?".

Research paper thumbnail of H-spectral spaces

Topology and Its Applications, 2006

Let X be a T 0 -space, we say that X is H -spectral if its T 0 -compactification is spectral. Thi... more Let X be a T 0 -space, we say that X is H -spectral if its T 0 -compactification is spectral. This paper deal with topological properties of H -spectral spaces. In the case of T 1 -spaces the T 0 -compactification coincides with the Wallman compactification. We give necessary and sufficient condition on the T 1 -space X in order to get its Wallman compactification spectral.

Research paper thumbnail of T(α,β)-spaces and the Wallman compactification

International Journal of Mathematics and Mathematical Sciences, 2004

Some new separation axioms are introduced and studied. We also deal with maps having an extension... more Some new separation axioms are introduced and studied. We also deal with maps having an extension to a homeomorphism between the Wallman compactifications of their domains and ranges.

Research paper thumbnail of Some Properties of Spectral Binary Relations

Research paper thumbnail of Submaximal and door compactifications

Topology and Its Applications, 2011

ABSTRACT In this paper, a characterization is given for compact door spaces. We, also, deal with ... more ABSTRACT In this paper, a characterization is given for compact door spaces. We, also, deal with spaces X such that a compactification K(X) of X is submaximal or door.Let X be a topological space and K(X) be a compactification of X.We prove, here, that K(X) is submaximal if and only if for each dense subset D of X, the following properties hold:(i)D is co-finite in K(X);(ii)for each x∈K(X)∖D, {x} is closed.If X is a noncompact space, then we show that K(X) is a door space if and only if X is a discrete space and K(X) is the one-point compactification of X.

Research paper thumbnail of Two classes of locally compact sober spaces

International Journal of Mathematics and Mathematical Sciences, 2005

We deal with two classes of locally compact sober spaces, namely, the class of locally spectral c... more We deal with two classes of locally compact sober spaces, namely, the class of locally spectral coherent spaces and the class of spaces in which every point has a closed spectral neighborhood (CSN-spaces, for short). We prove that locally spectral coherent spaces are precisely the coherent sober spaces with a basis of compact open sets. We also prove that CSN-spaces are exactly the locally spectral coherent spaces in which every compact open set has a compact closure.

Research paper thumbnail of A-spectral spaces

Topology and Its Applications, 2004

By an A-spectral space, we mean a topological space X such that the Alexandroff extension (one po... more By an A-spectral space, we mean a topological space X such that the Alexandroff extension (one point compactification) of X is a spectral space. We give necessary and sufficient conditions on the space X in order to get it A-spectral.

Research paper thumbnail of SUBMAXIMAL AND SPECTRAL SPACES

Mathematical Proceedings of The Royal Irish Academy, 2008

A topological space X is said to be submaximal if every dense subset of X is open. In this paper,... more A topological space X is said to be submaximal if every dense subset of X is open. In this paper, descriptions of submaximal spectral spaces and Stone submaximal spaces are given. Throughout this paper a number of illustrative examples are given.

Research paper thumbnail of On a conjecture about spectral sets

Topology and Its Applications, 2004

Let R be a commutative ring with identity. We denote by Spec(R) the set of prime ideals of R. Cal... more Let R be a commutative ring with identity. We denote by Spec(R) the set of prime ideals of R. Call a partial ordered set spectral if it is order isomorphic to (Spec(R), ⊆) for some R. A longstanding open question about spectral sets (since 1976), is that of Lewis and Ohm [Canad. J. Math. 28 (1976) 820, Question 3.4]: "If (X, ) is an ordered disjoint union of the posets (X λ , λ ), λ ∈ Λ, and if (X, ) is spectral, then are the (X λ , λ ) also spectral?".

Research paper thumbnail of H-spectral spaces

Topology and Its Applications, 2006

Let X be a T 0 -space, we say that X is H -spectral if its T 0 -compactification is spectral. Thi... more Let X be a T 0 -space, we say that X is H -spectral if its T 0 -compactification is spectral. This paper deal with topological properties of H -spectral spaces. In the case of T 1 -spaces the T 0 -compactification coincides with the Wallman compactification. We give necessary and sufficient condition on the T 1 -space X in order to get its Wallman compactification spectral.

Research paper thumbnail of T(α,β)-spaces and the Wallman compactification

International Journal of Mathematics and Mathematical Sciences, 2004

Some new separation axioms are introduced and studied. We also deal with maps having an extension... more Some new separation axioms are introduced and studied. We also deal with maps having an extension to a homeomorphism between the Wallman compactifications of their domains and ranges.

Research paper thumbnail of Some Properties of Spectral Binary Relations

Research paper thumbnail of Submaximal and door compactifications

Topology and Its Applications, 2011

ABSTRACT In this paper, a characterization is given for compact door spaces. We, also, deal with ... more ABSTRACT In this paper, a characterization is given for compact door spaces. We, also, deal with spaces X such that a compactification K(X) of X is submaximal or door.Let X be a topological space and K(X) be a compactification of X.We prove, here, that K(X) is submaximal if and only if for each dense subset D of X, the following properties hold:(i)D is co-finite in K(X);(ii)for each x∈K(X)∖D, {x} is closed.If X is a noncompact space, then we show that K(X) is a door space if and only if X is a discrete space and K(X) is the one-point compactification of X.

Research paper thumbnail of Two classes of locally compact sober spaces

International Journal of Mathematics and Mathematical Sciences, 2005

We deal with two classes of locally compact sober spaces, namely, the class of locally spectral c... more We deal with two classes of locally compact sober spaces, namely, the class of locally spectral coherent spaces and the class of spaces in which every point has a closed spectral neighborhood (CSN-spaces, for short). We prove that locally spectral coherent spaces are precisely the coherent sober spaces with a basis of compact open sets. We also prove that CSN-spaces are exactly the locally spectral coherent spaces in which every compact open set has a compact closure.

Research paper thumbnail of A-spectral spaces

Topology and Its Applications, 2004

By an A-spectral space, we mean a topological space X such that the Alexandroff extension (one po... more By an A-spectral space, we mean a topological space X such that the Alexandroff extension (one point compactification) of X is a spectral space. We give necessary and sufficient conditions on the space X in order to get it A-spectral.