Neighborhood spaces

Abstract

Neighborhood spaces, pretopological spaces, and closure spaces are topological space generalizations which can be characterized by means of their associated interior (or closure) operators. The category NBD of neighborhood spaces and continuous maps contains PRTOP as a bicoreflective subcategory and CLS as a bireflective subcategory, whereas TOP is bireflectively embedded in PRTOP and bicoreflectively embedded in CLS. Initial and final structures are described in these categories, and it is shown that the Tychonov theorem holds in all of them. In order to describe a successful convergence theory in NBD, it is necessary to replace filters by more generalp-stacks.

Weak and strong structures and the $${T_{3.5}}$$ T 3.5 property for generalized topological spaces

“…Initial, i.e., weak and final, i.e., strong structures for supratopological spaces -which are closely related to generalized topological spaces, cf. below -are proved to exist and are investigated in [26]. [6] required that a closure operator c : P (X) → P (X) should be increasing, and preserve finite unions, also called finitely additive, i.e., c∅ = ∅ (in [19] p. xiii groundedness) and A, B ⊂ X =⇒ c(A ∪ B) = (cA) ∪ (cB) (in [19] p. xiii and in [5], p. 65, Definition 6.1 additivity).…”

“…However, unfortunately the terminologies collide: categorical topologists used to call supratopological spaces also as closure spaces (cf. e.g., [26]), which is in conflict with the usage of the monograph [19]. We will use the term strong generalized topological space.…”

“…We say that A ⊂ P (X) is a stack (also called ascending) if A ∈ A and A ⊂ B ⊂ X imply B ∈ A (cf. [26]). For B ⊂ P (X) we write Stack B := {C ⊂ X | ∃B ∈ B such that C ⊃ B}.…”

ON $gW$-CONTINUOUS FUNCTIONS INDUCED BY GENERALIZED $w$-OPEN SETS IN $w$-SPACES

“…We introduced the weak neighborhood systems defined by using the notion of weak neighborhoods in [11]. The weak neighborhood system induces a weak neighborhood space which is independent of neighborhood spaces [4] and general topological spaces [2]. The notions of weak structure, w-space, W -continuity and W * -continuity were investigated in [12].…”

$WO$-CONTINUITY AND $WK$-CONTINUITY ON ASSOCIATED $w$-spaces

“…The author introduced the weak neighborhood systems defined by using the notion of weak neighborhoods in [13]. The weak neighborhood system induces a weak neighborhood space (briefly WNS) which is independent of neighborhood spaces [4] and general topological spaces [2]. In [13], the author introduced the notion of new interior operator and closure operator on a WNS.…”