@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 On A Bitopological (1,2)*- Proper Functions Sabiha I . Mahmood Department of Mathematics/College of Science/ Al-Mustansiriyah University Received in: 14 junuary 2013 Accepted in : 1 april 2013 Abstract In this paper, we introduce a new type of functions in bitopological spaces, namely, (1,2)*-proper functions. Also, we study the basic properties and characterizations of these functions . One of the most important of equivalent definitions to the (1,2)*-proper functions is given by using (1,2)*-cluster points of filters . Moreover we define and study (1,2)*-perfect functions and (1,2)*-compact functions in bitopological spaces and we study the relation between (1,2)*-proper functions and each of (1,2)*-closed functions , (1,2)*-perfect functions and (1,2)*-compact functions and we give an example when the converse may not be true . Key words: (1,2)*-proper functions, (1,2)*-perfect functions , (1,2)*-compact functions , (1,2)*-cluster points, (1,2)*- 2T -spaces , (1,2)*-compactly closed sets and (1,2)*-K-spaces . 358 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Introduction The concept of a bitopological space ),,X( 21 ττ was first introduced by Kelly [1] , where X is a nonempty set and 1τ , 2τ are topologies on X . Lellis Thivagar et. al. [2] introduced the concepts of (1,2)*-compact spaces and studied their properties . Ravi et. al. [3] introduced the concepts of (1,2)*-closed functions . The purpose of this paper is to introduce a new class of functions, namely , (1,2)*-proper functions . We give the definition by depending on the definition of (1,2)*-closed functions . Also, we study the characterizations and basic properties of (1,2)*-proper functions . We can prove that a (1,2)*-continuous function ),,Y(),,X(:f 2121 σσ→ττ is (1,2)*-proper if and only if whenever ξ is a filter on X and Yy ∈ is a (1,2)*-cluster point of )(f ξ , then there is a (1,2)*-cluster point x of ξ such that y)x(f = . Moreover we study (1,2)*-perfect functions and (1,2)*-compact functions in bitopological spaces and study the relation between (1,2)*-proper functions and each of (1,2)*-closed functions , (1,2)*-perfect functions and (1,2)*-compact functions and we give an example when the converse may not be true . Throughout this paper ),,X( 21 ττ , ),,Y( 21 σσ and ),,Z( 21 ηη (or simply X , Y and Z ) represent non-empty bitopological spaces on which no separation axioms are assumed, unless otherwise mentioned . If XA ⊆ , then ),,A( AA 21 ττ is called a bitopological subspace of ),,X( 21 ττ . 1.Preliminaries First we recall the following definitions: Definition(1.1)[4]: A subset A of a bitopological space ),,X( 21 ττ is called 21ττ -open if 21 UUA = where 11U τ∈ and 22U τ∈ . The complement of a 21ττ -open set is called 21ττ -closed . Notice that 21ττ - open sets need not necessarily form a topology [4] . Definition(1.2)[4]: Let A be a subset of a bitopological space ),,X( 21 ττ .Then: i) The 21ττ - closure of A , denoted by )A(cl21ττ , is defined by:- =ττ )A(cl21 }closedisF&FA:F{ 21 −ττ⊆ . ii) The 21ττ - interior of A , denoted by )Aint(21ττ , is defined by:- =ττ )Aint(21 }openisU&AU:U{ 21 −ττ⊆ . Definition(1.3)[2]: A function ),,Y(),,X(:f 2121 σσ→ττ from a bitopological space ),,X( 21 ττ into a bitopological space ),,Y( 21 σσ is called (1,2)*-continuous if )V(f 1− is 21ττ -closed set in X for every 21σσ -closed set V in Y . Definition(1.4)[5]: Let ∧∈αααα τ′τ ),,X( be a family of bitopological spaces . On the product set α ∧∈α π= XX we define a bitopological structure ),( τ′τ by taking τ as the product topology generated by the ατ s and τ′ as the product topology generated by the ατ′ s . 359 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Definition(1.5)[6]: A filter ξ on a set X is a non-empty collection of non-empty subsets of X which has the following properties:- i) Every finite intersection of sets of ξ belongs to ξ . ii) Every subset of X which contains a set of ξ belongs to ξ . Definition(1.6)[6]: A non-empty collection 0ξ of non-empty subsets of a set X is called a filter base for some filter on X if and only if for each 0 2 0 1 0 F,F ξ∈ , there exists 0 3 0F ξ∈ such that 20 1 0 3 0 FFF ⊆ . Definition(1.7):A subset A of a bitopological space ),,X( 21 ττ is called a (1,2)*-neighborhood of a point x in X if there exists a 21ττ -open set U in X such that AUx ⊆∈ . The family of all (1,2)*-neighborhoods of a point Xx ∈ is denoted by )x(N * . Definition(1.8): A filter ξ on a bitopological space ),,X( 21 ττ has Xx ∈ as an (1,2)*-cluster point (written x )*2,1( ∝ξ ) iff each ξ∈F meets each )x(NN ∗∈ . Remark(1.9): A filter ξ on a bitopological space ),,X( 21 ττ has Xx ∈ as an (1,2)*-cluster point iff }F:)F(cl{x 21 ξ∈ττ∈ . Proof: To prove that }.F:)F(cl{xx 21 )*2,1( ξ∈ττ∈⇔∝ξ  φ≠ξ∈∀∈∀⇔∝ξ∴ FN,F&)x(NNx * )*2,1(  ξ∈∀φ≠∈∀⇔ ∗ F,NF,)x(NN  ξ∈∀ττ∈⇔ F),F(clx 21 }.F:)F(cl{x 21 ξ∈ττ∈⇔  Definition(1.10): A filter base 0ξ on a bitopological space ),,X( 21 ττ has Xx ∈ as an (1,2)*- cluster point (written x )*2,1( 0 ∝ξ ) iff each 0F ξ∈ meets each )x(NN ∗∈ . Definition(1.11)[5]: A bitopological space ),,X( 21 ττ is called a (1,2)*- 2T -space (or quasi- Hausdorff space if for any two distinct points x and y of X , there are two 21ττ -open sets U and V such that Ux ∈ , Vy ∈ and φ=VU  . Definition(1.12)[2]: A bitopological space ),,X( 21 ττ is said to be a (1,2)*-compact space if and only if every 21ττ -open cover of X has a finite subcover . Theorem(1.13): A bitopological space ),,X( 21 ττ is (1,2)*-compact if and only if given any family ∧∈αα }F{ of 21ττ -closed subsets of X such that the intersection of any finite number of the αF is non-empty, then φ≠ ∧∈α αF . Proof: It is Obvious . 360 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Theorem(1.14): A bitopological space ),,X( 21 ττ is (1,2)*-compact if and only if every filter on X has an (1,2)*-cluster point . Proof: ⇒ Suppose that ),,X( 21 ττ is a (1,2)*-compact space and ξ be a filter on X . Take }F:)F(cl{ 21 ξ∈ττ=Ω , then by (1.5) Ω has the finite intersection property (f.i.p.) . But X is (1,2)*-compact , then by (1.13) , φ≠Ω . This implies that any point of this intersection is an (1,2)*-cluster point of ξ . ⇐ Let Ω be a collection of 21ττ -closed subsets of X such that Ω has the f.i.p . Then there exists a filter ξ on X such that ξ⊆Ω . ⇒ }A:)A(cl{}F:)F(cl{ 2121 Ω∈ττ⊆ξ∈ττ  . Since ξ has an (1,2)*-cluster point , then there exists }F:)F(cl{x 21 ξ∈ττ∈ ⇒ φ≠ξ∈ττ }F:)F(cl{ 21 . Since }A:)A(cl{}F:)F(cl{ 2121 Ω∈ττ⊆ξ∈ττ  , then φ≠Ω∈ττ }A:)A(cl{ 21 . But Ω is a collection of 21ττ -closed subsets of X , then }A:A{ Ω∈ }A:)A(cl{ 21 Ω∈ττ=  . Hence φ≠Ω∈ }A:A{ and by (1.13) , X is a (1,2)*-compact space . 2. (1,2)*-CLOSED FUNCTIONS Definition(2.1)[3]: A function ),,Y(),,X(:f 2121 σσ→ττ from a bitopological space ),,X( 21 ττ into a bitopological space ),,Y( 21 σσ is called (1,2)*-closed (resp. (1,2)*-open) if )F(f is 21σσ - closed (resp. 21σσ -open) in Y for every 21ττ -closed (resp. 21ττ -open) set F in X . Examples(2.2): i) Let ),,(),,(:f µµℜ→µµℜ be a function which is defined by : ℜ∈∀= x,0)x(f . Then f is an (1,2)*-closed function . ii) An inclusion function ),,X(),,F(:i FF τ′τ→τ′τ is (1,2)*-closed iff F is a τ′τ -closed set in X . Theorem(2.3): A function ),,Y(),,X(:f 2121 σσ→ττ is (1,2)*-closed if and only if for each subset B of Y and each 21ττ -open set U in X containing )B(f 1− , there exists a 21σσ -open set V in Y containing B such that U)V(f 1 ⊆− . Proof: ⇒ Suppose that B is an arbitrary subset in Y and U is an arbitrary 21ττ -open set in X containing )B(f 1− . Put )UX(fYV −−= . Then by ( 2.1) , V is a 21σσ -open set in Y . Since U)B(f 1 ⊆− ⇒ )BY(fUX 1 −⊆− − ⇒ BY)UX(f −⊆− ⇒ )UX(fYB −−⊆ ⇒ VB ⊆ and U)V(f 1 ⊆− . Conversely , Let F be any 21ττ -closed set in X . Put )F(fYB −= , then we have FX)B(f 1 −⊆− . Since FX − is 21ττ -open , then by hypothesis there exists a 21σσ -open set 361 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 V in Y such that VB ⊆ and FX)V(f 1 −⊆− . Therefore, we obtain VY)F(f −= and hence )F(f is 21σσ -closed in Y . This shows that f is a (1,2)*-closed function . Theorem(2.4): Let ),,X( 21 ττ , ),,Y( 21 σσ and ),,Z( 21 ηη be three bitopological spaces , and ),,Y(),,X(:f 2121 σσ→ττ , ),,Z(),,Y(:g 2121 ηη→σσ be two functions . Then:- i) If f and g are (1,2)*-closed , then fg  is (1,2)*-closed . ii) If fg  is (1,2)*-closed and f is (1,2)*-continuous and onto, then g is (1,2)*-closed . iii) If fg  is (1,2)*-closed and g is (1,2)*-continuous and one-to-one ,then f is (1,2)*-closed. Proof: i) To prove that ),,Z(),,X(:fg 2121 ηη→ττ is a (1,2)*-closed function . Let F be any 21ττ -closed subset of X . Since f is a (1,2)*-closed function, then )F(f is a 21σσ -closed set in Y . Since g is a (1,2)*-closed function , then ))F(f(g is a 21ηη -closed set in Z , hence )F)(fg(  is a 21ηη -closed set in Z . Thus ),,Z(),,X(:fg 2121 ηη→ττ is a (1,2)*-closed function . ii) To prove that ),,Z(),,Y(:g 2121 ηη→σσ is a (1,2)*-closed function . Let F be any 21σσ - closed subset of Y . Since f is (1,2)*-continuous, then )F(f 1− is a 21ττ -closed set in X . Since fg  is (1,2)*-closed, then ))F(ff(g))F(f)(fg( 11 −− =  is a 21ηη -closed set in Z . Since f is onto ,then )F(g is a 21ηη -closed set in Z . Thus ),,Z(),,Y(:g 2121 ηη→σσ is a (1,2)*-closed function . iii) To prove that ),,Y(),,X(:f 2121 σσ→ττ is a (1,2)*-closed function . Let F be any 21ττ - closed subset of X . Since fg  is (1,2)*-closed, then )F)(fg(  is a 21ηη -closed set in Z . Since g is (1,2)*-continuous, then ))F(f)(gg())F(fg(g 11  −− = is a 21σσ -closed set in Y . Since g is one-to-one, then )F(f is a 21σσ -closed set in Y . Thus ),,Y(),,X(:f 2121 σσ→ττ is a (1,2)*-closed function . Theorem(2.5): Let ),,Y(),,X(:f σ′σ→τ′τ be a (1,2)*-closed function . Then for each subset T of Y , the function ),,T(),),T(f(:f TT)T(f)T(f 1 T 11 σ′σ→τ′τ −− − which agrees with f on )T(f 1− is also (1,2)*-closed function . Proof: Let F be a )T(f)T(f 11 −− τ′τ -closed subset of )T(f 1− . Then there is a τ′τ -closed subset 1F of X such that )T(fFF 11 −=  . Since T)F(f)F(f 1T = and f is (1,2)*-closed function , then )F(f 1 is σ′σ -closed in Y . Hence T)F(f 1  is a TTσ′σ -closed set in T . Thus Tf is a (1,2)*- closed function . Theorem(2.6):If ),,Y(),,X(:f 21 σσ→τ′τ is a (1,2)*-closed function ,then the restriction of f to a τ′τ -closed subset F of X is a (1,2)*-closed function of F into Y . Proof: Since F is a τ′τ -closed set in X , then the inclusion function ),,X(),,F(:i FF τ′τ→τ′τ 362 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 is a (1,2)*-closed function . Since ),,Y(),,X(:f 21 σσ→τ′τ is a (1,2)*-closed function , then by (2.4) ),,Y(),,F(:if 21FF σσ→τ′τ is a (1,2)*-closed function . But F/fif = , thus the restriction function ),,Y(),,F(:F/f 21FF σσ→τ′τ is a (1,2)*-closed function . Remark(2.7): If ),,Y(),,X(:f 2112111 σσ→ττ and ),,Y(),,X(:f 2122122 σ′σ′→τ′τ′ are two (1,2)*-closed functions . Then ),,YY(),,XX(:ff 22112122112121 σ′×σσ′×σ×→τ′×ττ′×τ×× is not necessarily a (1,2)*-closed function . Example: Let ),,(),,(:f1 µµℜ→µµℜ be a function which is defined by : ℜ∈∀= x,0)x(f1 . And Let ),,(),,(:f 2 µµℜ→µµℜ be a function which is defined by : ℜ∈∀= x,x)x(f 2 . Where 2f is the identity function on ℜ . Clearly 1f and 2f are (1,2)*- closed functions , but ),,(),,(:ff 21 µ′µ′ℜ×ℜ→µ′µ′ℜ×ℜ× such that ℜ×ℜ∈∀=× )y,x(,)y,0()y,x)(ff( 21 (where µ′ is the product topology on ℜ×ℜ ) is not a (1,2)*-closed function , since the set }1yx:)y,x{(A =ℜ×ℜ∈= is µ′µ′ -closed in ℜ×ℜ , but }0{\}0{\}0{)A)(ff( 21 ℜ≅ℜ×=× is not µ′µ′ -closed in ℜ×ℜ . Theorem(2.8): Let ),,Y(),,X(:f 2112111 σσ→ττ and ),,Y(),,X(:f 2122122 σ′σ′→τ′τ′ be functions . If ),,YY(),,XX(:ff 22112122112121 σ′×σσ′×σ×→τ′×ττ′×τ×× is (1,2)*-closed , then 1f and 2f are also (1,2)*-closed functions . Proof: Suppose that ),,YY(),,XX(:ff 4321212121 ρρ×→ρρ×× is a (1,2)*-closed function where 4,3,2,1i,i =ρ be the product topology on 21 XX × and 21 YY × respectively . To prove that ),,Y(),,X(:f 2112111 σσ→ττ is (1,2)*-closed . Let F be a 21ττ -closed subset of 1X , to prove that )F(f1 is 21σσ -closed in 1Y . Suppose that )F(fG 1= ⇒ 2XF× is 21ρρ -closed in 21 XX × . Since 21 ff × is (1,2)*-closed ⇒ )X(fG)X(f)F(f)XF)(ff( 22221221 ×=×=×× is 43ρρ -closed in 21 YY × . i.e. )X(fG)X(fG(cl 222243 ×=×ρρ ⇒ G)G(cl21 =σσ ⇒ )F(fG 1= is 21σσ -closed in 1Y ⇒ ),,Y(),,X(:f 2112111 σσ→ττ is a (1,2)*-closed function . By the same way we can prove that 2f is a (1,2)*-closed function . Thus 1f and 2f are (1,2)*-closed functions . Definition(2.9)[7]: A function ),,Y(),,X(:f 2121 σσ→ττ from a bitopological space ),,X( 21 ττ into a bitopological space ),,Y( 21 σσ is called (1,2)*-homeomorphism if :- i) f is (1,2)*-continuous . ii) f is one-to-one and onto . iii) f is (1,2)*-closed (or (1,2)*-open ) . 3. (1,2)*-PROPER FUNCTIONS In this section we introduce a new type of functions in bitopological spaces which we call (1,2)*-proper functions . Besides we give examples and theorems . Definition(3.1): A function ),,Y(),,X(:f 2121 σσ→ττ from a bitopological space ),,X( 21 ττ into 363 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 a bitopological space ),,Y( 21 σσ is called (1,2)*-proper if :- i) f is (1,2)*-continuous . ii) ),,ZY(),,ZX(:If 2121Z τ′×στ×σ×→τ′×ττ×τ×× is (1,2)*-closed for every bitopological space ),,Z( τ′τ . Examples(3.2): i) Let ),,(),,(:f µµℜ→µµℜ be a function which is defined by : ℜ∈∀= x,0)x(f . Notice that f is a (1,2)*-closed function , but f is not a (1,2)*-proper function , since for the usual bitopological space ),,( µµℜ the function ),,(),,(:If µ×µµ×µℜ×ℜ→µ×µµ×µℜ×ℜ× ℜ which is defined by ℜ×ℜ∈∀=× ℜ )y,x(,)y,0()y,x)(If( is not a (1,2)*-closed function . ii) An inclusion function ),,X(),,F(:i FF τ′τ→τ′τ is (1,2)*-proper iff F is a τ′τ -closed set in X . Theorem(3.3):Every (1,2)*-proper function is a (1,2)*-closed function . Proof: Let ),,Y(),,X(:f 2121 σσ→ττ be a (1,2)*-proper function , then the function ),,ZY(),,ZX(:If 2121Z τ′×στ×σ×→τ′×ττ×τ×× is (1,2)*-closed for every bitopological space ),,Z( τ′τ . Let }t{Z = , then Y}t{YZY&X}t{XZX ≅×=×≅×=× and we can replace ZIf × by f . Thus ),,Y(),,X(:f 2121 σσ→ττ is a (1,2)*-closed function . Remark(3.4): The converse of (3.3) may not be true in general . Consider the following example: Example: In (3.2) (i) , ),,(),,(:f µµℜ→µµℜ is a (1,2)*-closed function, but it is not a (1,2)*- proper function . Theorem(3.5):Let ),,Y(),,X(:f 2121 σσ→ττ be a (1,2)*-continuous and one-to-one function. Then the following statements are equivalent:- i) f is (1,2)*-proper . ii) f is (1,2)*-closed . iii) f is a (1,2)*-homeomorphism of X onto a 21σσ -closed subset of Y . Proof: By theorem (3.3), )iii( → . )iiiii( → . Assume that ),,Y(),,X(:f 2121 σσ→ττ is a (1,2)*-closed function . Since X is a 21ττ -closed set in X , then )X(f is a 21σσ -closed set in Y . Since f is (1,2)*-continuous and one-to-one, then f is a (1,2)*-homeomorphism of X onto a 21σσ -closed subset )X(f of Y . )iiii( → . To prove that ),,ZY(),,ZX(:If 4321Z ρρ×→ρρ×× is (1,2)*-closed for every bitopological space ),,Z( τ′τ , where 4,3,2,1i,i =ρ be the product topology on ZX × and ZY × respectively . Since f is a (1,2)*-homeomorphism of X onto a 21σσ -closed subset F of Y , then ZIf × is a (1,2)*-homeomorphism of ZX × onto a 43ρρ -closed subset ZF× of 364 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 ZY × and therefore ZIf × is (1,2)*-closed . Thus ),,Y(),,X(:f 2121 σσ→ττ is a (1,2)*- proper function . Corollary(3.6): Every (1,2)*-homeomorphism is a (1,2)*-proper function . Remark(3.7): The converse of (3.6) may not be true in general . Consider the following example:- Example: Let ),,(),],1,0([:f µµℜ→µ′µ′ be a function which is defined by : ]1,0[x,x)x(f ∈∀= . (where µ′ is the relative usual topology on [0,1]) . Clearly that f is a (1,2)*-proper function, but it is not (1,2)*-homeomorphism . Theorem(3.8):Let ),,Y(),,X(:f σ′σ→τ′τ be a (1,2)*-proper function . Then for each subset T of Y , the function ),,T(),),T(f(:f TT)T(f)T(f 1 T 11 σ′σ→τ′τ −− − which agrees with f on )T(f 1− is also (1,2)*-proper . Proof: To prove that ),,T(),),T(f(:f TT)T(f)T(f 1 T 11 σ′σ→τ′τ −− − is (1,2)*-proper . Since f is (1,2)*-continuous , then so is Tf . Since f is (1,2)*-proper , then for every bitopological space ),,Z( τ′τ the function ),,ZY(),,ZX(:If 4321Z ρρ×→ρρ×× is (1,2)*-closed , where 4,3,2,1i,i =ρ be the product topology on ZX × and ZY × respectively . Since ZTZZT )If(If ××=× , then by (2.5) ZT If × is (1,2)*-closed . Thus ),,T(),),T(f(:f TT)T(f)T(f 1 T 11 σ′σ→τ′τ −− − is a (1,2)*-proper function . Definition(3.9): If the function ),,Y(),,X(:f 2121 σσ→ττ is (1,2)*-proper and ),,X( 21 ττ is a (1,2)*- 2T -space, then f is called a (1,2)*-perfect function . Corollary(3.10):Every (1,2)*-perfect function is a (1,2)*-proper function . Remark(3.11): The converse of (3.10) may not be true in general . Consider the following example:- Example: Let ),,(),,(:f .cof.cof.cof.cof ττℜ→ττℜ be the identity function , where .cofτ be the cofinite topology on ℜ . Then f is a (1,2)*-homeomorphism and by (3.6) , f is (1,2)*-proper . Since ),,( .cof.cof ττℜ is not a (1,2)*- 2T -space, then f is not a (1,2)*-perfect function . Theorem(3.12): Let ),,Y(),,X(:f 2121 σσ→ττ and ),,Z(),,Y(:g 2121 ηη→σσ be two (1,2)*- continuous functions . Then:- i) If f and g are (1,2)*-proper, then fg  is (1,2)*-proper . ii) If fg  is (1,2)*-proper and f is onto, then g is (1,2)*-proper . iii) If fg  is (1,2)*-proper and g is one-to-one , then f is (1,2)*-proper . 365 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Proof: i) It is clear that ),,Z(),,X(:fg 2121 ηη→ττ is (1,2)*-continuous . Let ),,Z( 1 τ′τ be any bitopological space . We have : )If()Ig(I)fg( !11 ZZZ ××=×  . Since f and g are (1,2)*- proper, then 1Z If × and 1Z Ig × are (1,2)*-closed . Hence by (2.4) , no.(i) !Z I)fg( × is (1,2)*- closed . Thus fg  is a (1,2)*-proper function . ii) To prove that ),,ZZ(),,ZY(:Ig 211211Z1 τ′×ητ×η×→τ′×στ×σ×× is (1,2)*-closed for every bitopological space 1Z . Since fg  is (1,2)*-proper, then )If()Ig(I)fg( !11 ZZZ ××=×  is (1,2)*-closed . Since f is (1,2)*-continuous and onto, then so is 1Z If × , hence by (2.4), no. (ii) 1Z Ig × is (1,2)*-closed . Thus ),,Z(),,Y(:g 2121 ηη→σσ is a (1,2)*-proper function . iii) To prove that ),,ZY(),,ZX(:If 211211Z1 τ′×στ×σ×→τ′×ττ×τ×× is (1,2)*-closed for every bitopological space 1Z . Since fg  is (1,2)*-proper, then )If()Ig(I)fg( !11 ZZZ ××=×  is (1,2)*-closed . Since g is one-to-one and (1,2)*-continuous, then so is 1Z Ig × , hence by (2.4) , no. (iii) 1Z If × is (1,2)*-closed . Thus ),,Y(),,X(:f 2121 σσ→ττ is a (1,2)*-proper function . Corollary(3.13): If ),,Y(),,X(:f 21 σσ→τ′τ is a (1,2)*-proper function, then the restriction of f to a τ′τ -closed subset F of X is a (1,2)*-proper function of F into Y . Proof: Since F is a τ′τ -closed set in X , then the inclusion function ),,X(),,F(:i FF τ′τ→τ′τ is a (1,2)*-proper function . Since ),,Y(),,X(:f 21 σσ→τ′τ is a (1,2)*-proper function , then by (3.12) , no.(i) ),,Y(),,F(:if 21FF σσ→τ′τ is a (1,2)*-proper function . But F/fif = , thus the restriction function ),,Y(),,F(:F/f 21FF σσ→τ′τ is a (1,2)*-proper function . Corollary(3.14): If ),,Y(),,X(:f 21 σσ→τ′τ is a (1,2)*-perfect function, then the restriction of f to a τ′τ -closed subset F of X is a (1,2)*-perfect function of F into Y . Proof: It is Obvious . Corollary(3.15): The composition of two (1,2)*-perfect functions is a (1,2)*-perfect function. Corollary(3.16):Let ),,Y(),,X(:f 2121 σσ→ττ and ),,Z(),,Y(:g 2121 ηη→σσ be functions . If f is a (1,2)*-perfect function and g is a (1,2)*-proper function . Then fg  is a (1,2)*- perfect function . Theorem(3.17): If ),,Y(),,X(:f 2112111 σσ→ττ and ),,Y(),,X(:f 2122122 σ′σ′→τ′τ′ are two (1,2)*-proper functions . Then ),,YY(),,XX(:ff 22112122112121 σ′×σσ′×σ×→τ′×ττ′×τ×× is also (1,2)*-proper function . Proof: Let ),,Z( τ′τ be any bitopological space .We can write Z21 Iff ×× by the composition of Z2Y IfI 1 ×× and ZX1 IIf 2 ×× . Since 1f and 2f are (1,2)*-proper functions, then 366 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 ZX1 IIf 2 ×× and Z2Y IfI 1 ×× are (1,2)*-closed functions , hence by (2.4), no. (i) )IIf()IfI( ZX1Z2Y 21 ××××  is (1,2)*-closed . But )IIf()IfI(Iff ZX1Z2YZ21 21 ××××=××  ⇒ Z21 Iff ×× is a (1,2)*-closed function . Thus ),,YY(),,XX(:ff 22112122112121 σ′×σσ′×σ×→τ′×ττ′×τ×× is a (1,2)*-proper function . Theorem(3.18): Let ),,Y(),,X(:f 2112111 σσ→ττ and ),,Y(),,X(:f 2122122 σ′σ′→τ′τ′ be two (1,2)*-continuous functions such that 21 ff × is a (1,2)*-proper function . Then , 1f and 2f are (1,2)*-proper . Proof: Let ),,Z( τ′τ be any bitopological space . Since 21 ff × is (1,2)*-proper , then ),,ZYY(),,ZXX(:Iff 43212121Z21 ρρ××→ρρ×××× is (1,2)*-closed , where 4,3,2,1i,i =ρ be the product topology on ZXX 21 ×× and respectively . To prove that ),,ZY(),,ZX(:If 432212Z2 ηη×→ηη×× is (1,2)*-closed , where 4,3,2,1i,i =η be the product topology on ZX 2 × and ZY2 × respectively . Let F be any 21ηη -closed set in ZX 2 × and )F)(If(G Z2 ×= . To prove that G is 43ηη -closed in ZY2 × . Since φ≠1X , then FX1 × is 21ρρ -closed in ZXX 21 ×× . Since Z21 Iff ×× is (1,2)*-closed, then G)X(f)FX)(Iff( 111Z21 ×=××× is 43ρρ -closed in ZYY 21 ×× ⇒ G)X(f)G)X(f(cl 111143 ×=×ρρ ⇒ G)G(cl43 =ηη ⇒ )F)(If(G Z2 ×= is 43ηη -closed in ZY2 × . Therefore Z2 If × is (1,2)*-closed . Thus 2f is a (1,2)*-proper function . By the same way we can prove that 1f is (1,2)*-proper . Lemma(3.19): Let ),,X( 21 ττ be any bitopological space such that the constant function }w{P),,X(:f 21 =→ττ is (1,2)*-proper .Then X is a (1,2)*-compact space, where w is any point which does not belong to X . Proof: Let ξ be a filter on X and let }w{XX =′ . Then }M:}w{M{ ξ∈=ξ′  is a filter on X′ . This filter with φ form a topology on X′ say τ . Hence ),,X( ττ′ is a bitopological space associated with ξ . Let XX ′×⊆∆ such that }Xx:)x,x{( ∈=∆ and let F)(cl21 =∆ρρ be the 21ρρ -closure of ∆ in ),,XX( 21 ρρ′× , where 2,1i,i =ρ be the product topology on XX ′× . Since P),,X(:f 21 →ττ is (1,2)*-proper , then XPXX:If X ′×→′×× ′ is (1,2)*-closed . But XXP ′≅′× so XXX:pr2 ′→′× is (1,2)*-closed . Hence )F(pr2 is ττ -closed in X′ . Since ∆∈)x,x( for each Xx ∈ ⇒ )(pr)x,x(prx 22 ∆∈= for each Xx ∈ ⇒ )F(prX 2⊆ ⇒ )F(pr))F(pr(cl)X(cl 22 =ττ⊆ττ . Since )X(clw ττ∈ ⇒ )F(prw 2∈ ⇒ Xx ∈∃ such that F)(cl)w,x( 21 =∆ρρ∈ .By the definition of the bitopology of XX ′× , this means that for each ZYY 21 ×× 367 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 (1,2)*-neighborhood V of x in X and each ξ∈M , we have φ≠∆× )MV( ⇒ φ≠MV  . Hence x is a (1,2)*-cluster point of the filter ξ . Thus X is a (1,2)*-compact space . Theorem(3.20): Let ),,Y(),,X(:f 2121 σσ→ττ be a (1,2)*-continuous function . Then the following statements are equivalent:- i) f is (1,2)*-proper . ii) f is (1,2)*-closed and )y(f 1− is (1,2)*-compact for each Yy ∈ . iii) If ξ is a filter on X and if Yy ∈ is a (1,2)*-cluster point of )(f ξ ,then there is a (1,2)*- cluster point x of ξ such that y)x(f = . Proof: )iii( → . If f is (1,2)*-proper ,then by (3.3) f is (1,2)*-closed . To prove that )y(f 1− is (1,2)*- compact for each Yy ∈ . Since f is (1,2)*-proper , then by (3.8) }y{)y(f:f 1}y{ → − is (1,2)*-proper for each Yy ∈ . By lemma (3.19) , we get )y(f 1− is (1,2)*-compact for each Yy ∈ . )iii( → . To prove that ),,ZY(),,ZX(:Ifh 4321Z ρρ×→ρρ××= is (1,2)*-closed for every bitopological space ),,Z( τ′τ , where 4,3,2,1i,i =ρ be the product topology on ZX × and ZY × respectively . Let C be any 21ρρ -closed in ZX × . To prove that D)C(h = is 43ρρ - closed in ZY × . Let cD)s,y( ∈ ⇒ )D(h)s,y(h c11 −− ∈ ⇒ )D(h)s,y()If( c11Z −− ∈× ⇒ )D(h)s,y)(If( c11Z 1 −−− ∈× ⇒ c1 C}s{)y(f ⊆×− , where cC is 21ρρ -open in ZX × ⇒ ∃ 21ττ -open set U in X and τ′τ -open set V in Z such that c1 CVU}s{)y(f ⊆×⊆×− ⇒ U)y(f 1 ⊆− and V}s{ ⊆ . Since f and ZI are (1,2)*-closed , then by (2.3) ∃ 21σσ -open set U′ in Y and τ′τ -open set V′ in Z such that U}y{ ′⊆ , V}s{ ′⊆ , U)U(f 1 ⊆′− and V)V(I 1Z ⊆′ − ⇒ cDVU)s,y( ⊆′×′∈ ⇒ cD is 43ρρ -open ⇒ D is 43ρρ -closed in ZY × . Hence ),,ZY(),,ZX(:If 4321Z ρρ×→ρρ×× is (1,2)*-closed . Thus ),,Y(),,X(:f 2121 σσ→ττ is (1,2)*-proper . )iiiii( → . Let ξ be a filter on X and Yy ∈ be a (1,2)*-cluster point of )(f ξ 368 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 ⇒ }F:))F(f(cl{y 21 ξ∈σσ∈ . Since f is (1,2)*-closed and (1,2)*-continuous , then by [1,8] ))F(cl(f))F(f(cl 2121 ττ=σσ for every ξ∈F ⇒ ))F(cl(fy 21ττ∈ for every ξ∈F ⇒ φ≠ττ− )F(cl)y(f 21 1  for every ξ∈F . Let }F:)F(cl)y(f{ 21 1 0 ξ∈ττ=ξ −  ⇒ 0ξ is a filter base on )y(f 1− whose elements are )y(f2)y(f1 11 −− ττ -closed subsets of ),),y(f( )y(f2)y(f1 1 11 −− ττ− . Since )y(f 1− is (1,2)*-compact , then by (1.14 ) there exist )y(fx 1−∈ such that }F:)F(cl)y(f{x 21 1 ξ∈ττ∈ −  ⇒ ∃ )y(fx 1−∈ such that )F(clx 21ττ∈ for every ξ∈F ⇒ x )*2,1( ∝ξ and y)x(f = . )iiiii( → . Let A be a non-empty 21ττ -closed subset of X and let ξ be the filter of subsets of X which contains A ⇒ A is the set of (1,2)*-cluster points of ξ . Let B be the set of (1,2)*- cluster points of )(f ξ on Y ⇒ B is 21σσ -closed set in Y and B)A(f ⊆ . To prove that )A(fB ⊆ , let By ∈ ⇒ y)(f )*2,1( ∝ξ by (iii), ∃ Xx ∈ such that x )*2,1( ∝ξ and y)x(f = . But A is the set of all (1,2)*-cluster points of ξ , then Ax ∈ and y)x(f = ⇒ )A(fy ∈ ⇒ )A(fB ⊆ ⇒ )A(fB = ⇒ f is (1,2)*-closed . Now , to prove that )y(f 1− is (1,2)*-compact for each Yy ∈ . Let Yy ∈ , then either φ=− )y(f 1 or φ≠− )y(f 1 . If φ=− )y(f 1 ⇒ )y(f 1− is (1,2)*-compact . If φ≠− )y(f 1 , then let ξ be a filter on )y(f 1− ⇒ )(f ξ be a filter generated by f on }y{ , but }y{ is (1,2)*-compact and }y{y ∈ , then y)(f )*2,1( ∝ξ in Y}y{ ⊆ . This implies that y)(f )*2,1( ∝ξ in Y . By (iii) ∃ Xx ∈ such that x )*2,1( ∝ξ and y)x(f = ⇒ )y(fx 1−∈ and )y(f 1− is (1,2)*- compact for each Yy ∈ . Corollary(3.21): A bitopological space ),,X( 21 ττ is (1,2)*-compact if and only if the constant function }w{P),,X(:f 21 =→ττ is (1,2)*-proper . Proof: It is Obvious . Theorem(3.22): If ),,X( 21 ττ is any (1,2)*-compact space and ),,Y( 21 σσ is any bitopological 369 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 space , then the projection ),,Y(),,YX(:pr 2122112 σσ→σ×τσ×τ× is a (1,2)*-proper function . Proof: Since ),,X( 21 ττ is a (1,2)*-compact space, then by (3.21) P),,X(:f 21 →ττ is (1,2)*- Proper . Since ),,Y(),,Y(:I 2121Y σσ→σσ is (1,2)*-proper , then by (3.17) YYPYX:If Y ≅×→×× is (1,2)*-proper . But Y2 Ifpr ×= . Thus ),,Y(),,YX(:pr 2122112 σσ→σ×τσ×τ× is a (1,2)*-proper function . Definition(3.23): A function ),,Y(),,X(:f 2121 σσ→ττ from a bitopological space ),,X( 21 ττ into a bitopological space ),,Y( 21 σσ is called (1,2)*-compact if the inverse image of every (1,2)*- compact set in Y is a (1,2)*-compact set in X . Theorem(3.24): Every (1,2)*-proper function is a (1,2)*-compact function . Proof: Let ),,Y(),,X(:f σ′σ→τ′τ be a (1,2)*-proper function and K be a (1,2)*- compact subset of Y, then by (3.8) , ),,K(),),K(f(:f KK)k(f)k(f 1 K 11 σ′σ→τ′τ −− − is (1,2)*-proper . Since PK → is (1,2)*-proper (by (3.21)) it follows from (3.12) , no.(i) that the composition PK)K(f Kf1 →→− is (1,2)*-proper . Hence by (3.21), )K(f 1− is (1,2)*-compact set in X . Remark(3.25): The converse of (3.24) may not be true in general .Consider the following example: Example: Let }c,b,a{YX == , }}a{,X,{1 φ=τ , }}c,b{,X,{2 φ=τ , }}a{,Y,{1 φ=σ and }}b{,Y,{2 φ=σ . The sets in }}c,b{},a{,X,{φ are 21ττ -closed sets in X and the sets in }}c{},c,a{},c,b{,Y,{φ are 21σσ - closed sets in Y . Let ),,Y(),,X(:f 2121 σσ→ττ be a function which is defined by : a)a(f = , b)b(f = and c)c(f = ⇒ f is a (1,2)*-compact function, but it is not (1,2)*-proper function ,since f is not (1,2)*-closed function . Theorem(3.26): Let ),,Y(),,X(:f 2121 σσ→ττ and ),,Z(),,Y(:g 2121 ηη→σσ be two (1,2)*- continuous functions . Then:- i) If f and g are (1,2)*-compact, then fg  is (1,2)*-compact . ii) If fg  is (1,2)*-compact and f is onto, then g is (1,2)*-compact . iii) If fg  is (1,2)*-compact and g is one-to-one , then f is (1,2)*-compact . Proof: The proof is similar of theorem (3.12) . 370 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Definition(3.27): A subset F of be a bitopological space ),,X( 21 ττ is said to be (1,2)*- compactly closed if KF  is (1,2)*-compact for each (1,2)*-compact set K in X . Remark(3.28): Every 21ττ -closed subset of a bitopological space ),,X( 21 ττ is (1,2)*- compactly closed . But the converse is not true in general . Consider the following example:- Example: Let }c,b,a{X = , }}a{,X,{1 φ=τ and }X,{2 φ=τ . The sets in }}c,b{,,X{ φ are 21ττ -closed . Thus }a{ is (1,2)*-compactly closed in X , but it is not 21ττ -closed . Definition(3.29): A (1,2)*- 2T -space ),,X( 21 ττ is called a (1,2)*-K-space if every (1,2)*- compactly closed subset of X is 21ττ -closed . Theorem(3.30): Let ),,Y(),,X(:f 2121 σσ→ττ be a (1,2)*-continuous function such that Y is a (1,2)*-K-space . Then f is a (1,2)*-proper function if and only if f is a (1,2)*-compact function . Proof: ⇒ By (3.24) every (1,2)*-proper function is a (1,2)*-compact function . Conversely , since f is a (1,2)*-compact function and }y{ is a (1,2)*-compact set in Y , then by (3.23) , )y(f 1− is (1,2)*-compact in X for each Yy ∈ . Now , to prove that f is (1,2)*-closed . Let F be any 21ττ -closed set in X , to prove that )F(f is a 21σσ -closed set in Y . Suppose that K is a (1,2)*-compact set in Y , then )K(f 1− is a (1,2)*-compact set in X . Since )K(fF 1− is a (1,2)*- compact set in X and f is (1,2)*-continuous, then by [5] ))K(fF(f 1− is a (1,2)*-compact set in Y . Since K)F(f))K(fF(f 1  =− , then by (3.27) )F(f is a (1,2)*-compactly closed set in Y . But Y is a (1,2)*-K-space , then by (3.29) )F(f is a 21σσ -closed set in Y . Therefore by (3.20) f is a (1,2)*-proper function . References 1. Kelly, J. C. (1963) Bitopological spaces , proc. London Math. Soc.,13 (3), 71-89 . 2. Lellis Thivagar, M.; Ravi, O. and Ekici, E. (2008) On (1,2)*-sets and decompositions of bitopological (1,2)*-continuous mappings , Kochi J. Math., 3 , 181-189. 3. Ravi, O. ; Jeyashri, S.; Pious Missier, S. and Nagendran, R. (2010) (1,2)*-semi-normal spaces and some bitopological functions (to appear) . 4. Ravi, O. and Lellis Thivagar, M. (2004) On stronger forms of (1,2)*- quotient mappings in bitopological spaces , Internat J. Math. Game Theory and Algebra , 14(6) , 481-492 .Datta, 371 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 5. M. C. (1972) Projective bitopological spaces , J. Austral. Math. Soc., 13 , 327–334. 6. Willard, S. (1970) General Topology , Addison-Wesley Inc., Mass . 7. Lellis Thivagar, M. ; Ravi, O. ; Joseph Israle, M. and Kayathri, K.(2009) decompositions of (1,2)*-rg-continuous maps in bitopological spaces (to appear in Antarctica J. Math.) . 8. Antony, J.; Ravi, O.; Pandi, A. and Santhana, C.M.(2011) On (1,2)*-s-normal spaces and pre-(1,2)*-gs-closed functions , International Journal of Algorithms ,Computing and mathematics, 4 (1) ,29-42 . 9. Ravi, O.; Pious Missier, S. and Salai Parkunan, T. (2011) On (1,2)*-semi-generalized-star Homeomorphisms,International Journal of Computer Science and Emerging Technologies , 2 (2) ,312-318 . 372 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 ) في الفضاءات التبولوجیة الثنائیة-*(1,2حول الدوال السدیدة صبیحة إبراھیم محمود الجامعة المستنصریة/ كلیة العلوم / قسم الریاضیات الخالصة -*(1,2جدیدا من الدوال في الفضاءات التبولوجیة الثنائیة أسمیناھا بالدوال السدیدة افي ھذا البحث قدمنا نوع . ((1,2)*-proper functions) احد أھم لتعارف-*(1,2كذلك درسنا الخواص األساسیة والمكافئات للدوال السدیدة . ( cluster points of-*(1,2)) . للمرشحات (*(1,2) -باستخدام النقاط العنقودیة الدوال أعطي المكافئة لھذه filters الدوال و *(1,2) (perfect functions-*(1,2)) -التامة الدوال درسنا و ذلك عرفنا عن فضال -*(1,2)تراصةالم ((1,2)*-compact functions) كذلك درسنا العالقة بین الدوال السدیدة .الفضاءات التبولوجیة الثنائیةفي- -والدوال المتراصة *(1,2)-والدوال التامة ( closed functions-*(1,2)) *(1,2)-الدوال المغلقة وكل من *(1,2) على التوالي مع أعطاء مثال لالتجاه غیر الصحیح. *(1,2) *(1,2) -النقاط العنقودیة ،-*(1,2)الدوال المتراصة ،)-*(1,2الدوال التامة ،)-*(1,2الدوال السدیدة المفتاحیة: الكلمات . *K -(1,2)-فضاءات ، *(1,2)-، المجموعات المغلقة رصا *2T-(1,2)-فضاءات ، 373 | Mathematics In this section we introduce a new type of functions in bitopological spaces which we call (1,2)*-proper functions . Besides we give examples and theorems . صبيحة إبراهيم محمود قسم الرياضيات / كلية العلوم / الجامعة المستنصرية الخلاصة