More on generalizations of topology of uniform convergence and m-topology on C(X)

Pratip Nandi ; Rakesh Bharati ; Atasi DebRay ; Sudip Kumar Acharyya

Filomat, Tome 37 (2023) no. 26, p. 8927

Résumé

This paper conglomerates our findings on the space C(X) of all real valued continuous functions, under different generalizations of the topology of uniform convergence and the m-topology. The paper begins with answering all the questions which were left open in our previous paper on the classifications of Z-ideals of C(X) induced by the U I and the m I-topologies on C(X) [5]. Motivated by the definition of the m I-topology, another generalization of the topology of uniform convergence, called U I-topology, is introduced here. Among several other results, it is established that for a convex ideal I in C(X), a necessary and sufficient condition for U I-topology to coincide with m I-topology on C(X) is the boundedness of X Z[I] in X. As opposed to the case of the U I-topologies (and m I-topologies) on C(X), it is proved that each U I-topology (respectively, m I-topology) on C(X) is uniquely determined by the ideal I. In the last section, the denseness of the set of units of C(X) in C U (X) (= C(X) with the topology of uniform convergence) is shown to be equivalent to the strong zero dimensionality of the space X. Also, the space X turns out to be a weakly P-space if and only if the set of zero divisors (including 0) in C(X) is closed in C U (X). Computing the closure of C P (X) (={ f ∈ C(X) : the support of f ∈ P} where P is an ideal of closed sets in X) in C U (X) and C m (X) (= C(X) with the m-topology), the results cl U C P (X) = C P ∞ (X) (= { f ∈ C(X) : ∀n ∈ N, {x ∈ X : | f (x)| ≥ 1 n } ∈ P}) and cl m C P (X) = { f ∈ C(X) : f. ∈ C P ∞ (X) for each ∈ C(X)} are achieved.

Détail

DOI : 10.2298/FIL2326927N

Classification : 54C40, 46E30
Keywords: Topology of uniform convergence, m-topology, UI, mI, UI and mI topologies, weakly P-space, strongly zero-dimensional space, CP(X), CP∞ (X)