By B. P. Dvalishvili

This monograph is the 1st and an preliminary creation to the idea of bitopological areas and its purposes. particularly, diversified households of subsets of bitopological areas are brought and numerous relatives among topologies are analyzed on one and an analogous set; the speculation of size of bitopological areas and the speculation of Baire bitopological areas are built, and numerous periods of mappings of bitopological areas are studied. The formerly recognized effects in addition the implications got during this monograph are utilized in research, power thought, normal topology, and idea of ordered topological areas. furthermore, a excessive point of recent wisdom of bitopological areas concept has made it attainable to introduce and examine algebra of latest variety, the corresponding illustration of which brings one to the exact category of bitopological areas. it truly is past any doubt that during the closest destiny the components of crucial purposes could be the theories of linear topological areas and topological teams, algebraic and differential topologies, the homotopy idea, let alone different primary components of contemporary arithmetic akin to geometry, mathematical common sense, the likelihood concept and lots of different components, together with these of utilized nature. Key good points: - First monograph is "Generalized Lattices" * the 1st creation to the speculation of bitopological areas and its functions.

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**Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures, and Applications**

This monograph is the 1st and an preliminary creation to the idea of bitopological areas and its functions. particularly, diversified households of subsets of bitopological areas are brought and diverse kinfolk among topologies are analyzed on one and an analogous set; the speculation of measurement of bitopological areas and the idea of Baire bitopological areas are built, and diverse sessions of mappings of bitopological areas are studied.

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**Additional info for Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures, and Applications**

**Example text**

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C ~ -,~ ~ ~ ~ ~ ~ ~ ~ ~ c~ ~ ~ ~ U ~-. ~ ~ ~ U ~. ~ ~ ~ "~" U ~ ~ c~ ~ U -~~ ~ ,,.. U ~. ~. ~ d ~ U ~ . ~ , I "~ U ~ ~. , I ~ U ~ ~ ~ I . ~ I II ~ ~ ~ ~. ~. ~. ~ . ,,i~ ~ ~ .. _- ~ "-' B~ ~o~. ~ II ~ ,~ ~ -~ "~ "--~ -~ "~ ,-. 5. 4. 2 cl Y. 2clY). ~ Proof. It suffices to show only the validity of the first implication. 1 cl A). ; cl(A n Y)). 2 cl (Y \ T; cl(A N Y)) - T2 cl (Y \ T;' cl(A N Y)). 3, we find that A n Y c (2, 1)-A/T)(Y). 3, we obtain the sufficient conditions for the relative (i, j)-nowhere densities.

A b~ ~ . Izr' ... ~ N ~ C ~ . '-" ' ~ 4g . il . ~ ~. ~ 49. o D ~. ~- ~. ~ ~ ~ " C -" C ~ k %. ~"-2. C qR. g. #. k ~ m - o.. U ~ II , u~ . II . U. : C~ U k- ~ ~U- ~ ~ ~U . ,~ 9~ ~ 0 4 ~ ~" F-1 c..? o . --2, b- ca ~ ~. ~ Ca ~ ~ "~ ~ ~ U 38 I. Different Families of Sets in Bitopological Spaces (4) if ~-j int Bt c C c Bt for some t E T, then C E 13. Then ( i , j ) - S C ( X ) C 13 and, therefore, ( i , j ) - $ C ( X ) is the smallest family of subsets of X satisfying (3) and (4). Proof. 16, the family ( i , j ) - S O ( X ) satisfies (1) and (2).