ISSN 2319-8133 (Online)
(An International Research Journal), www.compmath-journal.org
On Minimal Regular Open Sets and Maps in Topological Spaces
Anuradha N1 and Baby Chacko2
1Assistant Professor in Mathematics,
Government Engineering College, Kozhikode-5, Kerala, INDIA.
E-mail: [email protected].
2Associate Professor in Mathematics,
St. Joseph’s College, Devagiri, Kozhikode-8, Kerala, India.
E-mail: [email protected].
(Received on: April 1, 2015) ABSTRACT
In this paper a new class of topological spaces called rTmin spaces and rTmax spaces are introduced and relation between them is studied. Also new classes of maps called minimal r- continuous, maximal r-continuous, minimal r–irresolute, maximal r–irresolute, minimal-maximal r-continuous and maximal–minimal r- continuous maps in topological spaces are defined and relation of them with other types of functions is studied.
Keywords: Minimal regular open sets, Maximal regular open sets, minimal r–
continuous, maximal r–continuous.
1. INTRODUCTION
In the years 2001 and 2003, F. Nakaoka and N.Oda8,9,10 introduced and studied minimal open (resp. minimal closed) sets and maximal open (resp. maximal closed) sets, which are subclasses of open (resp. closed) sets. The complements of minimal open sets and maximal open sets are called maximal closed sets and minimal closed sets respectively. S.S.
Benchali, Basavaraj Ittanagi and R.S. Wali studied on Minimal open sets and maps in topological spaces2. Aim of this paper is to study minimal regular open sets and maps. A subset A of X is said to be regular open [Dugundji] if A= Int (Cl (A)) and regular closed if A= Cl (Int (A)).
1.1. Definition : A proper non empty open subset U of a topological space X is said to be a minimal open8 set, if any open set which is contained in U is φ or U.
1.2. Definition : A proper non empty regular open subset U of a topological space X is said to be a minimal regular open set, if any regular open set which is contained in U is φ or U.
1.3. Example : Let X={a, b, c} be with τ={φ, {a}, {b,c}, X}.Then {a} is a minimal regular open set.
1.4. Definition : A proper non empty open subset U of a topological space X is said to be a maximal open9 set, if any open set which contains U is X or U.
1.5. Definition : A proper non empty regular open subset U of a topological space X is said to be a maximal regular open set, if any regular open set which contains U is X or U.
1.6. Example : Let X={a, b, c} be with τ={φ,{a},{b,c}, X}. Then {b,c} is a maximal regular open set.
1.7. Definition : A proper non empty closed subset F of a topological space X is said to be a minimal closed10 set if any closed set which is contained in F is φ or F.
1.8. Definition : A proper non empty regular closed subset F of a topological space X is said to be a minimal regular closed set, if any regular closed set which is contained in F is φ or F.
1.9. Definition : A proper non empty closed subset F of a topological space X is said to be a maximal closed10 set, if any closed set which contains F is X or F.
1.10. Definition : A proper non empty regular closed subset F of a topological space X is said to be a maximal regular closed set, if any regular closed set which contains F is X or F.
1.11. Theorem : Let X be a topological space and F ⊂ X. F is a minimal regular closed set if and only if X−F is a maximal regular open set.
1.12. Theorem : Let X be a topological space and U ⊂ X. U is a minimal regular open set if and only if X−U is a maximal regular closed set.
1.13. Lemma :
(1) Let U be a minimal regular open set and W be a regular open set. Then U ∩ W = φ or U ⊂ W.
(2) Let U and V be minimal regular open sets. Then U ∩ V = φ or U = V.
1.14. Lemma :
(1) Let U be a maximal regular open set and W be a regular open set. Then U ∪ W = X or W ⊂ U.
(2) Let U and V be maximal regular open sets. Then U ∪ V = X or U = V.
1.15. Definition : A topological space (X, τ) is called
i) an r-door space if every subset is either regular open or regular closed in (X, τ).
ii) a sub maximal regular space if every r-dense subset of (X, τ) is regular open.
iii) an rT1/2 space if every regular closed subset of (X, τ) is closed in (X, τ).
1.16. Definition:
A map f : (X, τ) → (Y, σ) is called
i) almost continuous6 if f −1 (V) is an open set of (X, τ) for every regular open set V of (Y, σ).
ii) almost completely continuous4 if f −1 (V) is a regular open setof (X, τ) for every regular open set V of (Y, σ).
iii) δ-continuous12 if f −1 (V) is a δ-open set of (X, τ) for every regular open set V of (Y, σ).
iv) completely continuous1 if f −1(V) is aregular open set of (X, τ) for every open set V of (Y, σ).
v) strongly continuous7 if f −1 (V) is a clopen set of (X, τ) for every subset V of (Y, σ).
1.17. Definition : A topological space X is said to be i) δT0
4 if for each pair of distinct points x and y in X, there exists a regular open set containing one of the points x and y but not the other.
ii) δT1
4 if for each pair of distinct points x and y in X, there exist regular open sets U and V containing x and y respectively such that U contains x, but not y and V contains y but not x.
iii) rT2
1 if for each pair of distinct points x and y in X, there exists disjoint regular open sets U and V containing x and y respectively.
2. r Tmin AND rTmax SPACES
2.1 Definition : A topological space (X, τ) is said to be an rTmin space if every nonempty proper regular open subset of X is a minimal regular open set.
2.2 Definition : A topological space (X, τ) is said to be an rTmax space if every nonempty proper regular open subset of X is a maximal regular open set.
2.3 Remark : rTmin or rTmax topological spaces will be either discrete spaces or spaces of the form {φ, A, Ac, X}.
2.4. Theorem : A topological space (X,τ) is an rTmin space (resp. rTmax space) if and only if every non empty proper regular closed subset of X is a maximal regular closed (resp.
minimal regular closed) set in X.
Proof : The proof follows from the definition and the fact that complement of every minimal regular open (resp. maximal regular open) set is a maximal regular closed (resp. minimal regular closed) set.
2.5. Remark : Every pair of different minimal regular open (resp. maximal regular open) sets in an rTmin space (resp. rTmax space) are disjoint.
2.6. Remark : Union of every pair of different maximal regular open sets in an rTmax space is X.
2.7. Theorem : Let X be an rTmin space and Y be a regular open subspace of X. Then Y is also an rTmin space.
Proof : Let Y be a regular open subspace of an rTmin space X. Suppose U is not a minimal regular open set in Y. Then there exists a regular open set V≠ φ in Y such that V⊂U⊂Y.
Therefore V is a regular open set in Y which implies V is a regular open set X, which is contradiction to the fact that U is a minimal regular open set in X. Therefore U is a minimal regular open set in Y and Y is an rTmin space.
2.8. Remark : rTmin (resp.rTmax) and δT0 (resp. δT1, rT2) spaces are independent of each other.
Example:
i) X= {a, b, c}, τ = {φ, {a}, {b, c}, X}.
(X, τ) is an rTmin space (resp. rTmax) but it is not a δT0 (resp. δT1, rT2) space, since b, c
∈ X with b≠ c, there does not exist a regular open set containing b but not c or containing c but not b.
ii) Let X = {a, b, c} be with µ=P(X).Then X is not an rTmin space, but it is a δT0 (resp. δT1, rT2) space.
2.9. Remark: rTmin (resp. rTmax) space and rT1/2 space are independent of each other.
Example:
i) Let X = {a, b,c} with µ=P(X). Then X is not an rTmin (resp.rTmax) space, but it is an rT1/2
space.
ii) Let X ={a, b, c},τ={φ,{a},{b},{a, b},X}. Then X is rTmin (resp.rTmax) , but not rT1/2.
2.10. Remark : rTmin (resp.rTmax) space and r-door space are independent of each other.
Example:
i) X = {a ,b, c},τ={φ,{a},{b},{a, b},X}. Then X is not an r-door space, but it is an rTmin
space.
ii) X= {a,b,c}with µ=P(X). Then X is not an rTmin (resp.rTmax) space. But it is an r-doorspace.
2.11. Remark: rTmin (resp.rTmax) space and sub maximal spaces are independent of each other.
Example:
i) X = {a, b, c},τ={φ,{a},{b},{a, b},X}. Then X is not a sub maximal regular space. But it is an rTmin space.
ii) Let X = {a,b,c} be with µ=P(X). Then X is not an rTmin space. But it is a sub maximal regular space.
3. MINIMAL r-CONTINUOUS MAPS AND MAXIMAL r- CONTINUOUS MAPS
3.1. Definition : Let X and Y be topological spaces.
A map f : X→Y is called,
i) minimal r-continuous, if f –1 (M) is a regular open set in X, for every minimal regular open set M in Y.
ii) maximal r–continuous, if f –1 (M) is a regular open set in X, for every maximal regular open set M in Y.
iii) minimal r–irresolute, if f –1 (M) is a minimal regular open set in X, for every minimal regular open set M in Y.
iv) maximal r–irresolute if f –1 (M) is a maximal regular open set in X, for every maximal regular open set M in Y.
v) minimal–maximal r-continuous if f –1 (M) is a maximal regular open set in X, for every minimal regular open set M in Y.
vi) maximal–minimal r-continuous if f –1 (M) is a minimal regular open set in X, for every maximal regular open set M in Y.
3.2. Theorem: Every almost completely continuous function is minimal r-continuous.
3.3. Remark: Converse of the theorem need not be true.
3.4. Example: X=Y={a,b, c}, τ={φ,{a},{b},{a, b},{b, c},X}. Let f : X→Y be defined by f(a)=b, f(b)=a, f(c)=c. Then f is minimal r-continuous, but not almost completely continuous.
3.5. Theorem: If Y is an rTmin space and f: X→Y is minimal r-continuous onto map, then f is almost completely continuous.
3.6. Theorem: Every almost completely continuous map is maximal r-continuous.
3.7. Remark: Converse of the theorem need not be true.
3.8. Example: X=Y={a,b, c},τ={φ,{b},{a},{a, b},{b ,c},X}. Let f: X→Y be defined by f(a)=a, f(b)=c, f(c)=b. Then f is maximal r-continuous, but not almost completely continuous.
3.9. Theorem: Let f : X→Y be a maximal r-continuous onto map and let Y be an rTmax
space. Then f is almost completely continuous.
3.10. Theorem: Every strongly continuous function is minimal r-continuous.
Proof: Proof follows from the fact that regular open sets are open and clopen sets are regular open.
3.11. Remark: Converse of the theorem need not be true.
3.12. Example: Let X = Y={a ,b, c}, τ={φ,{a},{b},{a, b}, X} and µ ={φ, {a},{c}, {a, c}, Y}. Let f : X→Y be defined by f(a)=a, f(b)=c, f(c)=b. Then f : X→Y is minimal r- continuous, but not strongly continuous.
3.13. Theorem: If f : X→Y is minimal r-continuous, where Y is a locally indiscrete rTmin
space , X is locally indiscrete ,then f is strongly continuous.
3.14. Theorem: If f : X→Y is minimal r-continuous, where Y is an rTmin space, then f is almost continuous.
3.15. Theorem: If f : X→Y is almost continuous, where X is locally indiscrete, then f is minimal r-continuous.
3.16. Theorem: Every completely continuous function is minimal r-continuous.
3.17. Remark: Converse of the theorem need not be true.
3.18. Example: X=Y={a, b, c}.τ={φ,{a}, {b},{a, b}, X} and µ={φ, {a},{c}, {a, c}, Y}. Let f : X→Y be defined by f(a)=a, f(b)=c, f(c)=b Then f: X→Y is minimal r-continuous, but not completely continuous.
3.19. Theorem: Every minimal r-continuous function onto a locally indiscrete rTmin space is completely continuous.
3.20. Theorem: Every completely continuous is maximal r-continuous.
3.21. Remark: Converse of the above theorem need not be true.
3.22. Example: X=Y={a, b, c},τ={φ,{a}, {b},{a, b},X} and µ={φ,{a},{c},{a, c},Y}. Let f : X→Y be defined by f(a)=a, f(b)=c, f(c)=b Then f : X→Y is maximal r-continuous, but not
completely continuous.
3.23. Theorem: Every almost perfectly continuous function is minimal r-continuous.
3.24. Remark: Converse of the theorem need not be true.
3.25. Example: X = Y= {a, b, c}, τ={φ,{a},{b},{a, b},{b, c},X}. Let f : X→Y be defined by f(a)=b, f(b)=a, f(c)=c. Then f is minimal r-continuous, but not almost perfectly continuous.
3.26. Theorem: If f: X→Y is minimal r-continuous, where X is locally indiscrete and Y is rTmin , then f is almost perfectly continuous.
3.27. Theorem: If f: X→Y is maximal r-continuous, where X is locally indiscrete, Y is rTmax
space, then f is almost perfectly continuous.
3.28. Theorem: Every totally continuous function is minimal r-continuous.
3.29. Remark: Converse of the theorem need not be true.
3.30. Example: X=Y={a,b,c},τ ={φ,{a},{b},{a, b},{b, c},X}. Let f : X→Y be defined by f(a)=b, f(b)=a, (c)=c. Then f is minimal r-continuous, but not totally continuous.
3.31. Theorem: If f : X→Y is minimal r-continuous where X and Y are locally indiscrete and Y is rTmin , then f is totally continuous.
3.32. Theorem: Every totally continuous function is maximal r-continuous.
3.33. Remark: Converse of the above theorem need not be true.
3.34. Example: X=Y={a, b, c}, τ = {φ,{b},{a},{a, b},{b, c},X}. Let f : X→Y be defined by f(a)=a , f(b)=c , f(c)=b. Then f is maximal r-continuous, but not totally continuous.
3.35. Theorem: If f: X→Y is maximal r-continuous , where X and Y are locally indiscrete and Y is rTmax , then f is totally continuous.
3.36. Remark: Minimal r–continuous and maximal r–continuous maps are independent of each other.
3.37. Example:
i) X=Y={a ,b, c},τ={φ,{a},{b},{a ,b},{b ,c},X}. Let f: X→Y be defined by f(a)=b ,f(b)=a,
f(c)=c. Then f is minimal r–continuous, but not maximal r– continuous.
ii) X=Y={a, b, c},τ={φ,{b},{a},{a ,b},{b ,c}, X}. Let f : X→Y be defined by f(a)=a, f(b)=c, f(c)=b. Then f is maximal r-continuous, but not minimal r- continuous.
3.38. Theorem: Every minimal r–irresolute map is minimal r-continuous.
3.39. Remark: Converse of the theorem need not be true.
3.40. Example: X=Y={a,b,c}, τ={φ,{a},{b},{a, c},{b, c}{a, b},{c},X} and µ={φ,{a},{c}, {a,c},{b},{a,b},{b,c},Y}. Let f: X→Y be defined by f (a) =b, f(b)=b, f(c)=c. Then f is minimal r-continuous. But not minimal r-irresolute.
3.41. Theorem: Let f: X→Y be minimal r-continuous (resp. maximal r-continuous) where X is an rTmin space (resp.rTmax space). Then f is minimal r-irresolute (resp. maximal r- irresolute).
3.42. Theorem: Every minimal r-irresolute (resp.maximal r-irresolute) map onto an rTmin
space (resp.rTmax) is almost completely continuous.
3.43. Remark: Converse of the theorem need not be true.
3.44. Theorem: If X and Y are rTmin (resp.rTmax) spaces, every almost completely continuous function from X→Y is minimal r-irresolute (resp.maximal r-irresolute).
3.45. Theorem: Every minimal–maximal r-continuous map is minimal r-continuous.
3.46. Theorem: Every minimal r-continuous map from an rTmax space is minimal-maximal r- continuous.
3.47. Theorem: Every maximal-minimal r- continuous map is maximal r-continuous
3.48. Theorem: Every maximal r-continuous function from an rTmin space is maximal- minimal r-continuous.
3.49. Theorem: Let X and Y be topological spaces. A map f:X→Y is minimal r-continuous iff the inverse image of each maximal regular closed set in Y is a regular closed set in X.
3.50. Theorem: Let X and Y be topological spaces and A be a non empty regular open subset of X. If f : X→Y is minimal r-continuous, then the restriction map f/A: A→Y is minimal r-continuous.
3.51. Theorem: Let X and Y be topological spaces. A map f : X→Y is maximal r- continuous iff the inverse image of each minimal regular closed set in Y is a regular closed set in X.
3.52. Theorem: Let X and Y be topological spaces and A be a non empty regular open subset of X. If f: X→Y is maximal r-continuous, then the restriction map f/A: A→Y is
maximal r-continuous.
3.53. Theorem: Let X and Y be topological spaces. A map f : X→Y is minimal r-irresolute iff the inverse image of each maximal regular closed set in Y is maximal regular closed set in X.
3.54. Theorem: Let X & Y be topological spaces. A map f : X→Y is maximal r-irresolute iff the inverse image of each minimal regular closed set in Y is minimal regular closed set in X.
3.55. Theorem: Let X and Y be topological spaces. A map f : X→Y is maximal–minimal r- continuous iff the inverse image of each minimal regular closed set in Y is maximal regular closed set in X.
3.56. Remark : Compositions of minimal r-continuous functions need not be minimal r- continuous.
3.57. Example : X=Y= {a, b, c}, τ = {φ,{a},{b},{a, b},{b, c}, X}. Let f : X→Y be defined by f(a)=b, f(b)=c ,f(c)=c. Then f is minimal r-continuous. Let g: Y→X be defined by g(a)=b, g(b)=a, g(c)=c. Then g is minimal r-continuous. But g ◦ f : X→ X is not minimal r- continuous.
3.58. Theorem: If f : X→Y is almost completely continuous, g : Y→Z is minimal r- continuous , then g ◦ f : X→Z is minimal r-continuous.
3.59. Remark: Compositions of maximal r-continuous maps need not be maximal r- continuous.
Example:
X= {a, b, c},τ={φ,{b},{a},{a, b},{b ,c},X}. Let f : X→ X be defined by f(a)=b ,f(b)=a, f(c)=a. Then f is maximal r-continuous, but f ◦ f is not maximal r-continuous.
3.60. Theorem : If f : X→Y is almost completely continuous , g : Y→Z is maximal r- continuous , then g ◦ f : X→Z is maximal r-continuous.
3.61. Theorem : If f : X→Y and g : Y→Z are maximal r–irresolute maps, then g ◦ f : X→Z is maximal r-irresolute.
3.62. Remark: Composition of minimal –maximal r-continuous maps need not be minimal – maximal r-continuous.
3.63. Example: Let X=Y={a, b, c}, τ={φ,{b}{a, c},{a, b},{b, c},{c},X}. µ= {φ, {c}, {a, b},Y}. Suppose f : X→Y be defined by f(a)=b, f(b)=c, f(c)=a. Then f is minimal maximal r- continuous. But f ◦ f not minimal maximal r-continuous.
3.64. Theorem: If f : X→Y and g : Y→Z are minimal–maximal r-continuous and if Y is an rTmin space, then g ◦ f : X→Z is minimal maximal r–continuous.
3.65. Theorem: If f : X→Y is maximal r-irresolute and g : Y→Z is minimal maximal r-
continuous, then g ◦ f : X→Z is minimal maximal r-continuous.
3.66. Theorem: If f : X→Y is maximal r-continuous and g : Y→Z is minimal maximal r- continuous, then g ◦ f : X→Z is minimal r-continuous.
3.67. Theorem: Let X and Y be topological spaces. A map f : X→Y is maximal minimal r–
continuous iff the inverse image of each minimal regular closed set in Y is a maximal regular closed set in X.
3.68. Remark: Composition of maximal-minimal r-continuous maps need not be maximal minimal r–continuous.
3.69. Theorem: If f : X→Y is minimal r-irresolute and g:Y→Z is maximal– minimal r- continuous, then g ◦ f : X→Z is maximal minimal r-continuous.
3.70. Theorem: If f : X→Y is minimal r-continuous and g : Y→Z is maximal minimal r- continuous , then g ◦ f : X→Z is maximal r-continuous.
3.71. Theorem: If f : X→Y and g : Y→Z are maximal minimal r-continuous and if Y is an rTmax space,then g◦ f: X→Z is maximal minimal r-continuous.
3.72. Remark: From the above discussion and known results we have the following implications.
Minimal – maximal Maximal – minimal r-continuous r-continuous
Minimal r-continuous Almost completely Maximal r- continuous continuous
Minimal r–irresolute Maximal r–irresolute REFERENCES
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