Süleyman Önal

On characterization of a Riesz homomorphism on C(X)-space

Let X be a realcompact space. We present a very simple and elementary proof of the well known fact that every Riesz homomorphism π: C(X) → R is point evaluated. Moreover, the proof is given in ZF.

On weaker forms of the chain (F) condition and metacompactness-like covering properties in the product spaces

We introduce the concept of a family of sets generating another family. Then we prove that if X is a topological space and X has W = {W(x): x ∈ X} which is finitely generated by a countable family satisfying (F) which consists of families each Noetherian of ω-rank, then X is metaLindelöf as well as a countable product of them. We also prove that if W satisfies ω-rank (F) and, for every x ∈ X, W(x) is of the form W0(x) ∪ W1(x), where W0(x) is Noetherian and W1(x) consists of neighbourhoods of x, then X is metacompact.

An answer to a question on the affine bijections on C ( X,I )

A complete description of the bijective affine map on C(X, I) is given. This provides an answer to a question of [2] on the affine bijections on C(X, I).

Paracompactness of spaces which have covering properties weaker than paracompactness

Abstract We prove that (i) a collectionwise normal, orthocompact, θ m -refinable, [m,ℵ 0 ] -submetacompact space is paracompact, (ii) a collectionwise normal, [∞,m] -paracompact [m,ℵ 0 ] -submetacompact space is paracompact. This gives a sufficient condition for the paracompactness of para-Lindelof, collectionwise normal spaces.