The sum theorem and its corollaries are proved for a countable family of zero-dimensional (in the sense of small and large inductive bidimensions) p-closed sets, using a new notion of relative normality whose topological correspondent is also new. The notion of almost $n$-dimensionality is considered from the bitopological point of view. Bitopological spaces in which every subset is i-open in its $j$-closure (i.e.,(i,j)-submaximal spaces) are introduced and their properties are studied. Based on the investigations begun in [5] and [14], sufficient conditions are found for bitopological spaces to be(1,2)-Baire in the class of p-normal spaces. Furthermore, (i,j)-I-spaces are introduced and both the relations between(i,j)-submaximal, (i,j)-nodec and (i,j)-I-spaces, and their properties are studied when two topologies on a set are either independent of each other or interconnected by the inclusion, S-, C- and N-relations or by their combinations. The final part of the paper deals with the questions of preservation of $(i,j)$-submaximal and $(2,1)\dd I$-spaces to an image, of $D$-spaces to an image and an inverse image for both the topological and the bitopological cases. Two theorems are formulated containing, on the one hand, topological conditions and, on the other hand, bitopological ones, under which a topological space is a $D$-space.
Establishment of new results for topological as well as bitopological spaces and strengthening of certain existing and well-known ones have motivated this paper's systematic investigation of different classes of bitopological spaces.
All useful notions have been collected and the following abbreviations are used throughout the paper: TS for a topological space, TsS for a topological subspace, BS for a bitopological space and BsS for a bitopological subspace. The plural form of all abbreviations is ’s. Always i, j ∈ {1, 2}, i = j, unless stated otherwise.
Let (X, τ 1 , τ 2 ) be a BS and P be some topological property. Then (i, j)-P denotes the analogue of this property for τ i with respect to τ j , and p -P denotes the conjunction (1, 2)-P ∧ (2, 1)-P, that is, p -P denotes an “absolute” bitopological analogue of P, where “p” is the abbreviation for “pairwise”. Sometimes (1, 2)-P ⇐⇒ (2, 1)-P (and thus ⇐⇒ p -P) so that it suffices to consider one of these three bitopological analogues. Moreover, there are certain cases where equivalent topological formulations do not remain equivalent when passing to their bitopological counterparts; in particular, this phenomenon is observed in the case of submaximal spaces [7]. Also note that (X, τ i ) has a property P if and only if (X, τ 1 , τ 2 ) has a property i-P, and d-P is equivalent to 1-P ∧ 2-P, where “d” is the abbreviation for “double”.
If τ 1 and τ 2 are independent of each other on X, then along with the properties p -P and d-P we can cobsider the property sup P, where sup P is the P-property of the TS (X, sup(τ 1 , τ 2 )), clearly not be considered for the case τ 1 ⊂ τ 2 , that is, in our further notation, for a BS (X, τ 1 < τ 2 ).
The symbol 2 X is used for the power set of the set X, and for a family A = {A s } s∈S ⊂ 2 X , co A denotes the conjugate family {X \ A s : A s ∈ A} s∈S . If A ⊂ X, then τ i int A and τ i cl A denote respectively the interior and the closure of A in the topology τ i (for a TS (X, τ ) the closure of a subset A ⊂ X is denoted by
, where B i ∈ co τ i [11]. Thus, a subset
and the family of all p -open (p -closed) subsets of a BS (X, τ 1 , τ 2 ) is denoted by p -O(X) (p -Cl(X)). It is clear that τ 1 ∪ τ 2 ⊂ p -O(X) (co τ 1 ∪ co τ 2 ⊂ p -Cl(X)) and so in a BS (X, τ 1 < τ 2 ) we have p -O(X) = τ 2 (p -Cl(X) = co τ 2 ). The notion of a p -open (p -closed) set is equivalent to the notion of a quasi open (quasi closed) set given in [10]. The bitopological boundaries of a subset A ⊂ X are p -closed sets (i, j)-Fr A = τ i cl A ∩ τ j cl(X \ A) [11].
Also, to avoid confusion with generally accepted notations, for a BS (X, τ 1 , τ 2 ) we shall use the following double indexation:
x is an i-accumulation point of A and A i j = x ∈ X : x is a j-isolated point of A , that is, the lower indices i and j denote the belonging to the topology and, therefore, i, j ∈ {1, 2}, while the upper indices d and i are fixed as the accumulation and isolation symbols, respectively; thus A i j = A\A d j , A is a j-discrete set ⇐⇒ A = A i j , and
F n , F n ∈ co τ i for each n = 1, ∞ are the families of all i-boundary, i-dense, i-dense in themselves, (i, j)-dense in themselves, i-scattered, p -scattered, (i, j)-first category, (i, j)-second category, i-G δ and i-F σ -subsets of X, respectively; note also here, that a subset A of a BS (X, τ 1 , τ 2 ) is of (i, j)-first (second) category, i.e., A is of (i, j)-Catg I ((i, j)-Catg II) if it is of (i, j)-first (second) category in itself [15]. Definition 1.1. Let (X, τ 1 , τ 2 ) be a BS. Then (1) (X, τ 1 , τ 2 ) is R -p -T 1 (i.e., p -T 1 in the sense of Reilly) if it is d-T 1 [19].
(2) (X, τ 1 , τ 2 ) is (i, j)-regular if for each point x ∈ X and each i-closed set F ⊂ X, x ∈ F , there exist an i-open set U and a j-open set V that x ∈ U , F ⊂ V and U ∩ V = ∅ [16].
(3) (X, τ 1 , τ 2 ) is p -normal if for every pair of disjoint sets A, B in X, where A is 1-closed and B is 2-closed, there exist a 2-open set U and a 1-open set V such that A ⊂ U , B ⊂ V and U ∩ V = ∅ [16].
Moreover, (X, τ 1 , τ 2 ) is hereditarily p -normal if every one of its BsS is p -normal [11].
(4) (X, τ 1 , τ 2 ) is p -connected if X cannot be expressed as a union of two disjoint sets A and B such that A ∈ τ 1 \ {∅} and B ∈ τ 2 \ {∅} [18] (see also [6], [8], [17]).
(
) is an (i, j)-Baire space or an almost (i, j)-Baire space (briefly, (i, j)-BrS or A-(i, j)-BrS) if every nonempty i-open subset of X is of (i, j)-second category or of (i, j)-second category in X [13], [15].
(7) (X, τ 1 , τ 2 ) is (i, j)-nodec if its every (i, j)-nowhere dense subset is j-closed and i-discrete [14].
Furthermore, in a BS (X,
Since for a BS (X,
are correct, in the case where τ 1 ⊂ τ 2 we come to the following evident implications:
Moreover, according to (1) of Theorem 2.1.10 in [15], for a BS (X, τ 1 < S τ 2 ), where τ 1 < S τ 2 ⇐⇒ (τ 1 ⊂ τ 2 ∧ τ 1 Sτ 2 ), in addition to the above implications, we have:
The families of all such subsets of X are denoted by (i,
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