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Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(24): 2016 ๐น- Continuous in bi topological Space By Gem-set Iktifa Diaa Jaleel Open Educational CollegeMathematics Department , Iraq . [email protected] Abstract The study of a new type of continuity in bi topological space by Gem-set is introduced as . 1. Basic definition . 2. ๐ฟ- -continuous on (๐ฟ- -open , ๐ฟ- -closed , ๐ฟ- -interior , ๐ฟ- -closure) in bi topological space) , 3. ๐ฟ- -continuous on separation Axioms in bi topological space . with proof of some theorem . keywords: Gem-set, Bi topological, Theorem โซุงูุฎุงูุตุฉโฌ โซู ู ุฎุงูู ุฏุฑุงุณุชู ูู ูููู ((ู ุฌู ูุนุฉ ุงูุฌููุฑุฉ)) ุชูุตูุช ุงูู ููุน ุฌุฏูุฏ ู ู ุงุงูุณุชู ุงุฑุฑูุฉ ูู ุงููุถุงุกุงุช ุงูุซูุงุฆูุฉ ุงูุชุจูููุฌูุงโฌ โซุจูุงุณุทุฉ ุงูู ุฌู ูุนุฉ ุงูุฌููุฑุฉ ููุฏ ูู ุช ุจุฃุซุจุงุช ุจุนุถ ุงููุธุฑูุงุช ูุงูุฎุตุงุฆุต ุงูุฎุงุตุฉ ุจุงุงูุณุชู ุงุฑุฑูุฉ ููุฐูู ุจุฏูููุงุช ุงููุตู ุจุงุงูุนุชู ุงุฏ ุนููโฌ โซุงูู ุฌู ูุนุฉ ุงูุฌููุฑุฉโฌ .โซ ุงูุซูุงุฆูุฉ ุงูุชุจูููุฌูุง ูุธุฑูุฉโฌ,โซุงูุฌููุฑุฉ ุงููุถุงุกุงุชโฌ, โซ ุงูู ุฌู ูุนุฉโฌ:โซุงูููู ุงุช ุงูู ูุชุงุญูุฉโฌ Introduction This research establishes a relation between bi topological spaces , initiated by Kelly (1963) . Defined as : A set equipped with two topologies is called a bi topological space , denoted by (๐, ๐, ฮฉ) where (๐, ๐) , (๐, ฮฉ) are two topological space defined on ๐ฅ X and โ-Gem set in topological space , define ๐ดโ with respect to space (๐, ๐) as ๐ฅ follows ๐ดโ = {๐ฆ โ ๐; ๐บ โฉ ๐ด โ ฮ๐ฅ ๐๐๐ ๐๐ฃ๐๐๐ฆ ๐บ โ ๐(๐ฆ)} is called " Gem-set " in topological space . ๐ฅ A new definition for ๐ฟ- Gem-set in bi topological space define ๐ด = {๐ฆ โ ๐: ๐บ โฉ ๐ด โ ฮ๐ฅ , ๐๐๐ ๐๐ฃ๐๐๐ฆ ๐บ โ ๐(๐ฆ) ๐๐ ฮฉ(y)} from the relation above , the following generalization is formulated between โ- Gem-set in topological space and ๐ฟ -Gem-set in bi topological space . And the research consists of basic definition and ๐ฟ - continuous in bi topological space by Gem-sets (๐ฟ- -open , ๐ฟ- -closed , ๐ฟ- -interior , ๐ฟ- -closed) List of symbols Symbols ๐ฅ ๐ดโ ๐ฅ ๐๐โ (๐ด) ๐ด ๐ฅ ๐ฅ ๐๐ (๐ด) ๐(๐ฆ) Description โ-Gem set in (๐, ๐) ๐ฅ ๐ดโ โช ๐ด โ-Gem set in (๐, ๐, ฮฉ) ๐ด ๐ฅ โช๐ด The collection of all open subsets containing the point y 622 Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(24): 2016 ฮ๐ฅ I deal at point x ๐(๐ฅ) The neighborhood system at a point x in (๐, ๐) ๐(๐ฅ) The neighborhood system at a point x in (๐, ฮฉ) ๐ฟ- -cl(๐ฅ) The collection of all ๐ฟ -close subset in (๐, ๐, ฮฉ) ๐ฟ- -o(๐ฅ) The collection of all ๐ฟ -open subset in (๐, ๐, ฮฉ) ๐-int(๐ด) The set of all interior point of A in (๐, ๐) ฮฉ-cl(๐ด) The set of all closure subsets A of (๐, ฮฉ) ๐: ๐ฅ โ ๐ฆ Single-valued function 1. Basic Definitions 1.1 Definition by Kelly (1963): A set equipped with two topologies is called a bi topological space , denoted by (๐, ๐, ฮฉ) where (๐, ๐) , (๐, ฮฉ) are two topological spaces defined on X. 1.2 Definition by Noiril (1974) : Let (๐, ๐) be a topological space , and ๐ด โ ๐ , A is said to be โ-open set iff ๐ด โ ๐ด๐โ๐ . 1.3 Definition Let (๐, ๐, ฮฉ) be a bi topological space . and A be a subset of X A is said to be ๐ฟ-open set iff ๐ด โ ๐-int (ฮฉ-cl(๐-int(๐ด))) . 1) 1.4 Definition (Manoharan and Thangarelu , 2013) : Let X be a non-empty set , A family I of subset of X is an ideal on X if : i๐ด โ ๐ผ and ๐ต โ ๐ด , then ๐ต โ ๐ผ (heredity) ii๐ด, ๐ต โ ๐ผ , then ๐ด โช ๐ต โ ๐ผ (finite additivity) 1.5 Definition : Let (๐, ๐) be a topological space with an ideal I on X , Then for any subset A of X , ๐ดโ (๐ผ, ๐) = {๐ฅ โ ๐: ๐ โฉ ๐ด โ ๐ผ ๐๐๐ ๐๐ฃ๐๐๐ฆ ๐ โ ๐(๐ฅ)} is called the local function of A with respect to ๐ผ and ๐ , simply write ๐ดโ (๐ผ, ๐) is case then is no chance for confusion . Also , cl*(๐ด) = ๐ด โช ๐ดโ defines kuratowski closere operator for topology ๐ โ which is a finer then ๐ for a topological space (๐, ๐) and ๐ฅ โ ๐ , the ideal ๐ผ๐ฅ [25][4] define by ๐ผ๐ฅ = {๐บ โ ๐: ๐ฅ โ ๐บ ๐ } . 1.6 Definition(AL-Swidi and AL-Nafee ,2013) : ๐ฅ Let (๐, ๐) be a topological space , ๐ด โ ๐ and ๐ฅ โ ๐ Define ๐ดโ with respect to ๐ฅ ๐ฅ space (๐, ๐) as follows : ๐ดโ = {๐ฆ โ ๐: ๐บ โฉ ๐ด โ ๐ผ๐ฅ ๐๐๐ ๐๐ฃ๐๐๐ฆ ๐บ โ ๐(๐ฆ)} . A set ๐ดโ ๐ฅ ๐ฅ ๐ฅ is called "Gem-set" we write the ๐๐ โ (๐ด) = ๐ดโ โช ๐ด . Thus ๐๐ โ (๐น) โ ๐๐(๐น) A new definition for ๐ฟ- -open set in bi topological space. 623 Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(24): 2016 1.7 Definition : Let (๐, ๐, ฮฉ) be a bi topological space with an ideal I on X. for any subset A of X, ๐ด (๐ผ, ๐, ฮฉ) = {๐ฅ โ ๐; ๐ โฉ ๐ด โ ๐ผ ๐๐๐ ๐๐ฃ๐๐๐ฆ ๐ โ ๐(๐ฅ) ๐๐ (๐, ๐) ๐๐ ๐ โ ๐(๐ฅ) ๐๐ (๐, ฮฉ)} is called the local function of A with respect to I and ๐ or with respect to I and ฮฉ Also ฮฉ-cl (๐ด) = ๐ด โช ๐ด defines kuratowski chosere operator for ๐ โ is finer than ๐ . For a bi topological space (๐, ๐, ฮฉ) and ๐ฅ โ ๐ , the ideal ๐ผ๐ฅ define by ๐ผ๐ฅ = {๐บ โ ๐, ๐ฅ โ ๐บ ๐ } . 1.8 Definition : ๐ฅ Let (๐, ๐, ฮฉ) be a bi topological space , ๐ด โ ๐ and ๐ฅ โ ๐ , Define ๐ด with respet to space (๐, ๐, ฮฉ) as follows : ๐ฅ ๐ด = {๐ฆ โ ๐: ๐บ โฉ ๐ด โ ๐ผ๐ฅ , ๐๐๐ ๐๐ฃ๐๐๐ฆ ๐บ โ ๐(๐ฆ) ๐๐ ๐บ โ ฮฉ(๐ฆ)} ๐ฅ ๐ฅ ๐ฅ A set ๐ด is called "Gem-set" . we write the ๐๐ (๐ด) = ๐ด โช ๐ด Thus ๐ฅ ๐๐ (๐น) โ ฮฉ-cl(๐น) . 1.9 Definition A subset A of an bi topological space (๐, ๐, ฮฉ) is called i. ๐ฟ-open if ๐ด โ ๐-int(ฮฉ-cl(๐-int(A))) ii. ๐ฟ-open if ๐ด โ ๐-int(ฮฉ-cl (๐-int(A))) The collection of all ๐ฟ-open sets in (ii) is denoted by ๐ฟ- -o(๐ฅ) and the collection of all ๐ฟ-open sets is denoted by ๐ฟ-o(๐ฅ) 1.10 Definition A bi topological space (๐, ๐, ฮฉ) is said to be ๐ฟ- -closed space if and only if each non-empty subset A of X is ๐ฟ- -closed subset . 1.11 Definition A bi topological space (๐, ๐, ฮฉ) is said to be ๐ฟ- -perfected space if and only if each non-empty subset A of X is ๐ฟ- - perfected subset . 1.12 Lemma Let (๐, ๐, ฮฉ) be a bi topological space with I and J being ideal son X, and let A and B be two subset X then i๐ด โ ๐ต then ๐ด โ ๐ต ii๐ผ โ ๐ฝ then ๐ด (๐ฝ) โ ๐ต (๐ผ) iii๐ด = ฮฉ-cl(๐ด ) โ ฮฉ- cl(๐ด) iv(๐ด ) โ ๐ด (๐ด โช ๐ต) = ๐ด โช ๐ต vvi๐ด โ ๐ต = (๐ด โ ๐ต) โ ๐ต โ (๐ด โ ๐ต) viiFor every ๐ผ1 โ ๐ผ , (๐ด โ ๐ผ1 ) = (๐ด โ ๐ผ) If this is casy to prove by the properties in Lemma (1.12) with respect to Gem-sets . 1.13 Definition Let (๐, ๐, ฮฉ) be a bi topological space and ๐ด โ ๐ , defined ๐๐ ๐ด , for each ๐ฅ โ ๐ . 624 ๐ฅ (๐ด) = ๐ด ๐ฅ โช Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(24): 2016 1.14 Definition A subset A of a bi topological space (๐, ๐, ฮฉ) is called perfected set if ๐ด ๐ด , for each ๐ฅ โ ๐ . ๐ฅ โ 1.15 definition A bi topological space (๐, ๐, ฮฉ) is said to be perfected space if and only if , each non-empty subset A if y is perfected subset . 2. ๐น- - continuous map in bi topological space 2.1 ๐น- - continuous on (๐น- - open , ๐น- - closed , ๐น- -interior , ๐น- closure) in bi topological space 2.1.1 Definition Let (๐, ๐, ฮฉ) and (๐, ๐ฬ , ฮฉฬ) be a bi topological space , A mapping ๐: ๐ โถ ๐ is said to be ๐ฟ- -continuous at ๐ฅโ โ ๐ ๐๐๐ for every ๐ฟ- -open set V in Y containing ๐(๐ฅโ ) there exist ๐ฟ- -open set U in Y containing ๐ฅโ such that ๐(๐) โ ๐ . 2.1.2 Definition : Let ๐: ๐ โถ ๐ be a mapping , then (a) ๐ is said to be ๐ฟ- -open mapping ๐๐๐ ๐(๐บ) is ๐ฟ- -open in Y for every ๐ฟ- -open set G in X . (b) F is ๐ฟ- -closed ๐๐๐ ๐(๐น) is ๐ฟ- -closed in Y for every ๐ฟ- -closed set F in X . (c) ๐ is ๐ฟ- -continuous ๐๐๐ ๐ is ๐ฟ- -open and ๐ฟ- -closed . (d) ๐ is ๐ฟ- -homeomorphism iff i๐ is bijective (1-1 , onto) ii๐ and ๐ โ1 are ๐ฟ- -continuous where (๐, ๐, ฮฉ) , (๐, ๐ฬ , ฮฉฬ) are two bi topological space . 2.1.3 Example (1) Let ๐ = {๐, ๐, ๐} , ๐ = {โ , ๐, {๐}} , ฮฉ = {โ , ๐} (๐, ๐) , (๐, ฮฉ) are two topologies on X Then (๐, ๐, ฮฉ) is a bi topological space , such that ๐ฟ- -o(๐ฅ) = {ฮฆ, ๐, {๐}, {๐, ๐}, {๐, ๐}} And let ๐ = {1,2,3} , ๐ฬ = {โ , ๐, {1}} , ฮฉฬ = {โ , ๐} (๐, ๐ฬ ) , (๐, ฮฉฬ) are two topologies on ๐ Then (๐, ๐ฬ , ฮฉฬ) is a bi topological space , such that ๐ฟ- -o(๐ฆ) = {โ , ๐, {1}, {1,2}, {1,3}} Define ๐: (๐, ๐, ฮฉ) โ (๐, ๐ฬ , ฮฉฬ) by ๐(๐) = 1, ๐(๐) = 2, ๐(๐) = 3 Then ๐ is ๐ฟ- -continuous and ๐ฟ- -open set because ๐ โ1 (๐) = {๐, ๐, ๐} = ๐ is ๐ฟ- -open in ๐ , and ๐ โ1 (โ ) = โ is ๐ฟ- -open in ๐ , similarly the other cases ๐ โ1 ({1}), ๐ โ1 ({1,2}), ๐ โ1 ({1,3}) are ๐ฟ- -open in ๐ , there fore ๐ is ๐ฟ- -continuous . and since ๐({๐}) = {1} is ๐ฟ- -open in ๐ and ๐({โ }) = โ is ๐ฟ- -open in ๐ , similarly the other cases ๐({๐ฅ}) , ๐({๐, ๐}) , ๐({๐, ๐}) are ๐ฟ- -open in ๐ . therefore ๐ is ๐ฟ- -open mapping . 2.1.3 Example (2) Let ๐ = {๐, ๐, ๐} , ๐ = {โ , ๐, {๐}} , ฮฉ = {โ , ๐, {๐}, {๐, ๐}} 625 Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(24): 2016 (๐, ๐) , (๐, ฮฉ) are two topologies on X Then (๐, ๐, ฮฉ) is a bi topological space , such that ๐ฟ- -o(๐ฅ) = {โ , ๐, {๐}} And let ๐ = {1,2,3} , ๐ฬ = {โ , ๐, {2}, {1}, {1,2}} , ฮฉฬ = {โ , ๐, {1}, {1,2}} (๐, ๐ฬ ) , (๐, ฮฉฬ) are two topologies on ๐ Then (๐, ๐ฬ , ฮฉฬ) is a bi topological space , such that ๐ฟ- -o(๐ฆ) = {โ , ๐, {1}, {2}, {1,2}} Define ๐: (๐, ๐, ฮฉ) โ (๐, ๐ฬ , ฮฉฬ) by ๐(๐) = 1, ๐(๐) = ๐(๐) = 2 Then ๐ is ๐ฟ- -open but not ๐ฟ- -continuous because ๐ โ1 (๐) = {๐, ๐, ๐} = ๐ is ๐ฟ- -open in ๐ and ๐ โ1 (โ ) = โ is ๐ฟ- -open in ๐ . โ1 ({2}) but ๐ = {๐, ๐} is not ๐ฟ- -open in ๐ , Hence ๐ is ๐ฟ- -continuous . And since ๐(๐ฅ) = {1,2} is ๐ฟ- -open in ๐ , ๐({โ }) = โ is ๐ฟ- -open in ๐ . ๐(๐) = ({1}) is ๐ฟ- -open in ๐ therefore ๐ is ๐ฟ- -open . 2.1.5 Theorem Let (๐, ๐, ฮฉ) and (๐, ๐ฬ , ฮฉฬ) be bi topological space , then a mapping ๐: ๐ โ ๐ is ๐ฟ- -continuous ๐๐๐ for every ๐ฅ โ ๐ the inverse image under ๐ of every ๐ฟ- -open ๐ โ ๐ ๐(๐ฅ) is ๐ฟ- -open set of ๐ . Proof Let ๐ is ๐ฟ- '-continuous and ๐ is ๐ฟ- -open in ๐ to prove ๐ โ1 (๐) is ๐ฟ- open in ๐ . If ๐ โ1 (๐) = โ so it is ๐ฟ- -open in ๐ . If ๐ โ1 (๐) โ โ , Let ๐ฅ โ ๐ โ1 (๐) , then ๐(๐ฅ) โ ๐ , by definition of ๐ฟ- -continuous there exists ๐ฟ- -open ๐บ๐ฅ in ๐ containing ๐ฅ such that ๐(๐บ๐ฅ ) โ ๐ . โด ๐ฅ โ ๐บ๐ฅ โ ๐ โ1 (๐) , Hence ๐ โ1 (๐) is ๐ฟ- -open set in ๐ . Conversely Let ๐ โ1 (๐) is ๐ฟ- -open set in ๐ , for each ๐ is ๐ฟ- -open set in ๐ to prove ๐ is ๐ฟ- -continuous . Let ๐ฅ โ ๐ and ๐ is ๐ฟ- -open set in ๐ containing ๐(๐ฅ) so ๐ โ1 (๐) is ๐ฟ-open in ๐ containing ๐ฅ and ๐(๐ โ1 (๐)) โ ๐ . Then ๐ is ๐ฟ- -continuous on ๐ . 2.1.6 Theorem Let (๐, ๐, ฮฉ) and (๐, ๐ฬ , ฮฉฬ) be bi topological space , a mapping ๐: ๐ โ ๐ is ๐ฟ-continuous ๐๐๐ the inverse image under ๐ โ ๐ every ๐ฟ- -closed set in ๐ is ๐ฟ- closed set in ๐ . Proof : (Obvious) 2.1.7 Theorem A mapping ๐: ๐ โ ๐ is ๐ฟ- -continuous ๐๐๐ ๐(๐ฟ- -cl(๐ด))โ ๐ฟ- -cl(๐(๐ด)) for every ๐ด โ ๐ , where (๐, ๐, ฮฉ) and (๐, ๐ฬ , ฮฉฬ) are two bi topological space . Proof Let ๐ be ๐ฟ- -continuous . Since ๐ฟ- -cl(๐(๐ด)) is ๐ฟ- -closed set in ๐ โด ๐ โ1 (๐ฟ-cl(๐(๐ด))) is ๐ฟ- -closed set in ๐ by [2.1.6] therefore ๐ฟ- -cl(๐ โ1 (๐ฟ- -cl(๐(๐ด))) = ๐ โ1 (๐ฟ- -cl(๐(๐ด)))โฆ(1) Now ๐(๐ด) โ ๐ฟ-cl(๐(๐ด)) , ๐ด โ ๐ โ1 (๐(๐ด)) โ ๐ โ1(๐ฟ- -cl(๐(๐ด))) . Then ๐ฟ- -cl(๐ด) โ ๐ โ1(๐ฟ- -cl(๐(๐ด)))= ๐ โ1(๐ฟ- -cl(๐(๐ด)))by1 626 Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(24): 2016 Then ๐(๐ฟ- -cl(๐ด)) โ ๐ฟ- -cl(๐(๐ด)) . Conversely : Let ๐(๐ฟ- -cl(๐ด)) โ ๐ฟ- -cl(๐(๐ด)) for every ๐ด โ ๐ Let ๐น be any ๐ฟ- -closed set in ๐ , So that ๐ฟ-cl(๐น) = ๐น Now , ๐ โ1 (๐น) โ ๐ , by hypothesis . ๐(๐ฟ- -cl(๐ โ1 (๐น)))โ ๐ฟ- -cl(๐(๐ โ1 (๐น))) โ ๐ฟ- -cl(๐น) = ๐น Therefore ๐ฟ- -cl(๐ โ1 (๐น)) โ ๐ โ1 (๐น) But ๐ โ1 (๐น) โ ๐ฟ- -cl(๐ โ1 (๐น)) always Hence ๐ฟ-cl(๐ โ1 (๐น)) โ ๐ โ1 (๐น) and ๐ โ1 (๐น) are ๐ฟ- -closed set in ๐ Hence ๐ is ๐ฟ- continuous by theorem [2.1.6] 2.1.8 Theorem A mapping ๐: ๐ โ ๐ is ๐ฟ- -continuous ๐๐๐ . ๐ฟ- -cl(๐ โ1 (๐ต)) โ ๐ โ1 (๐ฟ- cl(๐ต)) for every ๐ต โ ๐ , Where (๐, ๐, ฮฉ) and (๐, ๐ฬ , ฮฉฬ) are two bi topological space . Proof : (Obvious) 2.1.9 Theorem A mapping ๐: ๐ โ ๐ is ๐ฟ- -continuous ๐๐๐ ๐ โ1 (๐ฟ- -int(๐ต)) โ ๐ฟ- -int(๐ โ1 (๐ต)) for every ๐ต โ ๐ , where (๐, ๐, ฮฉ) and (๐, ๐ฬ , ฮฉฬ) are two bi topological space . Proof : (Obvious) 2.1.10 Theorem Let ๐, ๐ and ๐ be a bi topological space and the mappings ๐: ๐ โ ๐ and ๐: ๐ โ ๐ be ๐ฟ- -continuous then the composition map ๐ โ ๐: ๐ โ ๐ is ๐ฟ- continuous . Proof : (Obvious) (using definition 2.1.2 c+d) 2.2 ๐น- -continuous on separation Axioms in Bi topological space 2.2.1 Theorem Let (๐, ๐ฬ , ฮฉฬ) be ๐ฟ- -To space , if ๐: (๐, ๐, ฮฉ) โ (๐, ๐ฬ , ฮฉฬ) is ๐ฟ- -continuous 1-1 function . Then (๐, ๐, ฮฉ) is ๐ฟ- -To space . Proof : Let ๐ฅ1 , ๐ฅ2 โ ๐ , ๐ฅ1 โ ๐ฅ2 , since ๐ is 1-1 function , then ๐(๐ฅ1 ) โ ๐(๐ฅ2 ) , ๐(๐ฅ2 ) โ ๐ , and ๐ is ๐ฟ- -To space , then there exists ๐ฟ- -open set ๐บ in ๐ such that ๐(๐ฅ1 ) โ ๐บ , ๐(๐ฅ1 ) โ ๐บ So ๐ฅ1 โ ๐ โ1 (๐บ) , ๐ฅ2 โ ๐ โ1 (๐บ) . โด ๐ โ1 (๐บ) is ๐ฟ- -open set in ๐ , Then (๐, ๐, ฮฉ) is ๐ฟ- -To space . 2.2.2 Theorem Let ๐: (๐, ๐, ฮฉ) โ (๐, ๐ฬ , ฮฉฬ) be an ๐ฟ- -continuous ๐ฟ- -open 1-1 and onto function , If (๐, ๐, ฮฉ) is ๐ฟ-To space then (๐, ๐ฬ , ฮฉฬ) is ๐ฟ- -To space . Proof : Suppose that ๐ฆ1 , ๐ฆ2 โ ๐ , ๐ฆ1 โ ๐ฆ2 , since f is onto , there exists ๐ฅ1 , ๐ฅ2 โ ๐ , such that ๐ฆ1 = ๐(๐ฅ1 ) , ๐ฆ2 = ๐(๐ฅ2 ) and since ๐ is 1-1 , then ๐ฅ1 โ ๐ฅ2 , since ๐ is ๐ฟ-To space . There exists ๐ฟ- -open set ๐บ , such that ๐ฅ1 โ ๐บ , ๐ฅ2 โ ๐บ . Hence ๐ฆ1 = ๐(๐ฅ1 ) โ ๐(๐บ) , ๐ฆ2 = ๐(๐ฅ2 ) โ ๐(๐บ) , since ๐ is ๐ฟ- -open function , then ๐(๐บ) is ๐ฟ- -open set ๐ . there fore (๐, ๐ฬ , ฮฉฬ) is ๐ฟ- -To space . 627 Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(24): 2016 2.2.3 Theorem Let (๐, ๐ฬ , ฮฉฬ) be ๐ฟ-T1 space . if ๐: (๐, ๐, ฮฉ) โ (๐, ๐ฬ , ฮฉฬ) is ๐ฟ- -continuous 1-1 function , then ๐ is ๐ฟ- -T1 space . Proof Let ๐ฅ1 , ๐ฅ2 โ ๐ , ๐ฅ1 โ ๐ฅ2 , since ๐ is 1-1 , ๐(๐ฅ1 ) โ ๐(๐ฅ2 ) , ๐(๐ฅ1 ) , ๐(๐ฅ2 ) โ ๐ , ๐ is ๐ฟ- -T1 space , then there exists ๐1 , ๐2 ๐ฟ- -open set in ๐ such that ๐(๐ฅ1 ) โ ๐1 , but ๐(๐ฅ2 ) โ ๐1 and ๐(๐ฅ2 ) โ ๐2 but ๐(๐ฅ1 ) โ ๐2 . Then ๐ฅ1 โ ๐ โ1 (๐1 )but ๐ฅ2 โ ๐ โ1 (๐1 ) ; and ๐ฅ2 โ ๐ โ1 (๐2 ), but ๐ฅ1 โ ๐ โ1 (๐2 ); โ1 (๐ ) โ1 and ๐ 1 , ๐ (๐2 ) are ๐ฟ- -open set in ๐ Hence (๐, ๐, ฮฉ) is ๐ฟ- -T1 space . 2.2.4 Theorem Let ๐: (๐, ๐, ฮฉ) โ (๐, ๐ฬ , ฮฉฬ) be an ๐ฟ- -continuous 1-1 and onto , ๐ฟ- -open function . If (๐, ๐, ฮฉ) is ๐ฟ- -T1 space then (๐, ๐ฬ , ฮฉฬ) is ๐ฟ- -T1 space . Proof Suppose ๐ฆ1 , ๐ฆ2 โ ๐ , ๐ฆ1 โ ๐ฆ2 , since ๐ is onto , there exists ๐ฅ1 , ๐ฅ2 โ ๐ , Such that ๐ฆ1 = ๐(๐ฅ1 ) , ๐ฆ2 = ๐(๐ฅ2 ) , since ๐ is 1-1 then ๐ฅ1 โ ๐ฅ2 โ ๐ , ๐(๐ฅ1 ) โ ๐(๐ฅ2 ) , and ๐ is ๐ฟ- -T1 space , there exists ๐ฟ- -open sets ๐บ , ๐ป such that ๐ฅ1 โ ๐บ but ๐ฅ2 โ ๐บ and ๐ฅ2 โ ๐ป but ๐ฅ1 โ ๐ป . Hence ๐(๐ฅ1 ) โ ๐(๐บ) , ๐(๐ฅ2 ) โ ๐(๐ป) , since ๐ is ๐ฟ- -open function , Hence ๐(๐บ) , ๐(๐ป) are ๐ฟ- -open sets of ๐ . ๐ฆ1 โ ๐(๐บ) , but ๐ฆ2 โ ๐(๐บ) and ๐ฆ2 โ ๐(๐ป) , but ๐ฆ1 โ ๐(๐ป) Then (๐, ๐ฬ , ฮฉฬ) is ๐ฟ- -T1 space . 2.2.5 theorem Let (๐, ๐ฬ , ฮฉฬ) be ๐ฟ- -T2 space . if ๐: (๐, ๐, ฮฉ) โ (๐, ๐ฬ , ฮฉฬ) is ๐ฟ- -continuous 11 function , then (๐, ๐, ฮฉ) is ๐ฟ- -T2 space . Proof Let ๐ฅ1 โ ๐ฅ2 โ ๐ , since ๐ is 1-1 , ๐(๐ฅ1 ) โ ๐(๐ฅ2 ) Let ๐ฆ1 = ๐(๐ฅ1 ) , ๐ฆ2 = ๐(๐ฅ2 ) , ๐ฆ1 โ ๐ฆ2 . since ๐ is ๐ฟ- -T2 space , there exists two ๐ฟ- -open sets ๐บ . ๐ป in ๐ , such that ๐ฆ1 โ ๐บ , ๐ฆ2 โ ๐ป , ๐บ โฉ ๐ป = โ . Hence ๐ฅ1 โ ๐ โ1 (๐บ) , ๐ฅ2 โ ๐ โ1 (๐ป) since ๐ is ๐ฟ- -continuous and โ1 (๐บ) ๐ , ๐ โ1 (๐ป) ๐ฟ- -open sets in ๐ . Also ๐ โ1 (๐บ) โฉ ๐ โ1 (๐ป) = ๐ โ1 (๐บ โฉ ๐ป) = ๐ โ1 (โ ) = โ Thus (๐, ๐, ฮฉ) is ๐ฟ- -T2 space . 2.2.6 Theorem Let ๐: (๐, ๐, ฮฉ) โ (๐, ๐ฬ , ฮฉฬ) be an ๐ฟ- -continuous 1-1 and onto , ๐ฟ- -open function . If (๐, ๐, ฮฉ) is ๐ฟ- -T2 space then (๐, ๐ฬ , ฮฉฬ) is ๐ฟ- -T2 space . Proof Let ๐ฆ1 โ ๐ฆ2 โ ๐ , since ๐ is 1-1 and onto , there exists ๐ฅ1 โ ๐ฅ2 โ ๐ , Such that ๐ฆ1 = ๐(๐ฅ1 ) , ๐ฆ2 = ๐(๐ฅ2 ) , since ๐ is ๐ฟ- -T2 space , then there exists ๐ฟ- -open sets ๐บ, ๐ป such that ๐ฅ1 โ ๐บ , ๐ฅ2 โ ๐ป , ๐บ โฉ ๐ป = โ , since ๐ is ๐ฟ- -open mapping , then ๐(๐บ) , ๐(๐ป) are two ๐ฟ- -open set in ๐ and ๐(๐บ โฉ ๐ป) = ๐(๐บ) โฉ ๐(๐ป) = ๐(โ ) = โ . Also ๐ฆ1 = ๐(๐ฅ1 ) โ ๐(๐บ) , ๐ฆ2 = ๐(๐ฅ2 ) โ ๐(๐ป) . Hence (๐, ๐ฬ , ฮฉฬ) is ๐ฟ- -T2 space . 2.2.7 Theorem Let (๐, ๐, ฮฉ) be ๐ฟ- -regular space and ๐: (๐, ๐, ฮฉ) โ (๐, ๐ฬ , ฮฉฬ) be ๐ฟ- -homeomorphism , Then (๐, ๐ฬ , ฮฉฬ) ๐ฟ- -regular. 628 Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(24): 2016 Proof Let ๐น be ๐ฟ- -closed set in ๐ , ๐ โ ๐น , ๐ โ ๐ . since ๐ is 1-1 and onto map , then there exists ๐ โ ๐ such that ๐(๐) = ๐ , ๐ = ๐ โ1 (๐) . since ๐ is ๐ฟ- -continuous so ๐ โ1 (๐น) is ๐ฟ- -closed in ๐ , ๐ โ ๐น , ๐ = ๐ โ1 (๐) โ ๐ โ1 (๐น) . since (๐, ๐, ฮฉ) is ๐ฟ- regular there exists ๐ฟ- -open sets ๐บ, ๐ป such that ๐ โ ๐บ , ๐ โ1 (๐น) โ ๐ป and ๐บ โฉ ๐ป = โ . So ๐ = ๐(๐) โ ๐(๐บ) , ๐น โ ๐(๐ โ1 (๐น)) โ ๐(๐ป) , since ๐ is ๐ฟ- -open map , hence ๐(๐บ) , ๐(๐ป) are ๐ฟ- -open sets in ๐ and ๐(๐บ โฉ ๐ป) = ๐(๐บ) โฉ ๐(๐ป) = ๐(โ ) = โ . Therefore (๐, ๐ฬ , ฮฉฬ) is ๐ฟ- -regular . 2.2.8 Theorem ๐ฟ- -normality is bi topological property . Proof Let (๐, ๐, ฮฉ) be ๐ฟ- -normal space and Let (๐, ๐ฬ , ฮฉฬ) be ๐ฟ- -homeomorphic image of (๐, ๐, ฮฉ) under ๐ฟ- -homeomorphic ๐ to show that (๐, ๐ฬ , ฮฉฬ) is also . ๐ฟ- normal space . Let ๐ฟ, ๐ be a pair of dis joint ๐ฟ- -closed subsets of ๐ . since ๐ is ๐ฟ- continuous map , then ๐ โ1 (๐ฟ) and ๐ โ1 (๐) are ๐ is ๐ฟ- -closed subsets of ๐ . Also ๐ โ1 (๐ฟ) โฉ ๐ โ1 (๐) = ๐ โ1 (๐ฟ โฉ ๐) = ๐ โ1 (โ ) = โ . Thus ๐ โ1 (๐ฟ), ๐ โ1 (๐) are disjoint pair of ๐ฟ- -closed subsets of ๐ . since the space (๐, ๐, ฮฉ) is ๐ฟ- -normal , then there exist ๐ฟ- -open set ๐บ and ๐ป such that ๐ โ1 (๐ฟ) โ ๐บ , ๐ โ1 (๐) โ ๐ป and ๐บ โฉ ๐ป = โ but ๐ โ1 (๐ฟ) โ ๐บ then ๐(๐ โ1 (๐ฟ)) โ ๐(๐บ) , ๐ฟ โ ๐(๐บ) . Similarly ๐ โ ๐(๐ป) , Also since ๐ is an ๐ฟ- -open mapping ๐(๐บ) and ๐(๐ป) are ๐ฟ- open subset of ๐ , such that ๐(๐บ) โฉ ๐(๐ป) = ๐(๐บ โฉ ๐ป) = ๐(โ ) = โ Thus there exists ๐ฟ-open subset in ๐ , ๐บ1 = ๐(๐บ) and ๐ป1 = ๐(๐ป) such that ๐ฟ โ ๐บ1 , ๐ โ ๐ป1 , and ๐บ1 โฉ ๐ป1 = โ . It follows that (๐, ๐ฬ , ฮฉฬ) is also ๐ฟ- -normal space . Accordingly , ๐ฟ- -normality is a bi topological property . References AL-Swidi L.A. and AL-Nafee A.B. "New separation Axioms using the ideal of "Gem-set" in topological space "Mathematical Theory and Moodeling , vol.3(3) (2013) , pp 60 โ 66 . Kelly,J.L. , (1963) "General Topology" , Van. Nostrands Princeton . Manoharan R. and Thangarelu , P. "Some New sets and Topologies in Ideal Topological space" , chine Journal of Mathematics . Article ID 973608 , pp. 1-6 , (2013) Noiril , T. ; Smashhour, A. ; Khedr F.H.; and A. Hsaanbi N. 1974 . 629