Abstract
In this paper, p-topologicalness (a relative topologicalness) in ⊤-convergence spaces are studied through two equivalent approaches. One approach generalizes the Fischer’s diagonal condition, the other approach extends the Gähler’s neighborhood condition. Then the relationships between p-topologicalness in ⊤-convergence spaces and p-topologicalness in stratified L-generalized convergence spaces are established. Furthermore, the lower and upper p-topological modifications in ⊤-convergence spaces are also defined and discussed. In particular, it is proved that the lower (resp., upper) p-topological modification behaves reasonably well relative to final (resp., initial) structures.
Highlights
The theory of convergence spaces [1] is natural extension of the theory of topological spaces.The topologicalness is important in the theory of convergence spaces since it mainly researches the condition of a convergence space to be a topological space
L-generalized convergence spaces was studied by Jäger [27,28,29] and Li [30,31], the p-topologicalness and p-topological modifications in stratified L-generalized convergence spaces were discussed by Li [32,33]
We say a pair of >-convergence spaces ( X, p, q) satisfy the Fischer >-diagonal condition if p p-(TF): Let J, X be any sets, ψ : J −→ X, and φ : J −→ F>
Summary
The theory of convergence spaces [1] is natural extension of the theory of topological spaces. When p = q, p-topologicalness is equivalent to topologicalness in convergence spaces They defined and discussed the lower and upper p-topological modifications in convergence spaces. For a pair of convergence spaces ( X, p) and ( X, q), the lower (resp., upper) p-topological modification of ( X, q) is defined as the finest (resp., coarsest) p-topological convergence space which is coarser (resp., finer) than ( X, q). L-generalized convergence spaces was studied by Jäger [27,28,29] and Li [30,31], the p-topologicalness and p-topological modifications in stratified L-generalized convergence spaces were discussed by Li [32,33]. The lower and upper p-topological modifications in >-convergence spaces are defined and discussed. It is proved that the lower (resp., upper) p-topological modification behaves reasonably well relative to final (resp., initial) structures
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