Download νg-open Mappings - Serials Publications

International Journal of Computer, Mathematical Sciences and Applications
Vol. 5, No. 1-2, January-June 2011, pp. 7– 14
© Serials Publications
ISSN: 0973-6786
g-open Mappings
S. BALASUBRAMANIAN
Department of Mathematics, Government Arts College (Autonomous), Karur (T.N.) India.
E-mail: [email protected]
Abstract: The aim of the paper is to study basic properties of νg-open mappings and interrelations
with other mappings.
AMS-classification Numbers: 54C10; 54C08; 54C05
Keywords: νg-open mappings.
1. INTRODUCTION
Mappings place an important role in the study of modern mathematics, especially in Topology and
Functional Analysis. Open mappings are one such mappings which are studied for different types of
open sets by various mathematicians for the past many years. Norman levine introduced the notion
of generalized open sets. After him different mathematicians worked and studied on different versions
of generalized open sets and related topological properties. In this paper we are going to further
study weak form of open mappings namely νg-open mappings using νg-open sets. Basic properties
are verified. Throughout the paper X, Y means a topological spaces (X, τ) and (Y, σ) unless otherwise
mentioned without any separation axioms.
2. PRELIMINARIES
Definition 2.1: A ⊂ X is said to be
(i) regular open[pre-open; semi-open; α-open; β-open] if A = ( A)° [ A ⊆ ( A)°; A ⊆ ( A°);
(
)
(
)
A ⊆ ( A°) °; A ⊆ ( A )°  and regular closed[pre-closed; semi-closed; α-closed; β-closed] if A =

A°  ( A° ) ⊆ A; ( A )° ⊆ A; ( ( A)° ) ⊆ A; ( A° ) ° ⊆ A]
(
)
(ii) ν-open[rα-open] if there exists a regular open set O such that O ⊂ A ⊂ O O ⊂ A ⊂ α (O) 
(iii) g-closed[resp: rg-closed] if A ⊆ U whenever A ⊆ U and U is open[resp: regular open].
(iv) αg-closed[resp: gα-closed; rgα-closed] if α ( A) ⊆ U whenever A ⊆ U and U is open[resp:
α-open; rα-open].
(v) gs-closed[resp: sg-closed] if s ( A) ⊆ U whenever A ⊆ U and U is open[resp: semi-open].
(vi) gp-closed[resp: pg-closed; gpr-closed; pgpr-closed; sgp-closed] if p ( A) ⊆ U whenever
g-open; semi-open].
A ⊆ U and U is open[resp: pre-open; regular-open; rg
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International Journal of Computer, Mathematical Sciences and Applications
(vii) gsp-closed[resp: pre-semi-closed] if s( A) ⊆ U whenever A ⊆ U and U is open[resp:
g-open].
(viii) νg-closed if ν( A) ⊆ U whenever A ⊆ U and U is ν-open in X.
Note 1: From the above definition we have the following implication diagram.
Definition 2.2: A function f : X → Y is said to be
1. continuous[resp: semi-continuous; nearly-continuous; ν-continuous; g-continuous;
rg-continuous] if the inverse image of every open set is open[resp: semi-open; regular-open;
ν-open; g-open; rg-open]
2. irresolute[resp: nearly-irresolute; ν-irresolute] if the inverse image of every semi-open[resp:
regular-open; ν-open] set is semi-open[resp: regular-open; ν-open]
3. open[resp: semi-open; nearly-open] if image of every open set is open[resp: semi-open;
regular-open]


Definition 2.3: X is said to be T1  r − T1  if every [regular-]generalized closed set is
2 
2
[regular-]closed
3.
g-OPEN MAPPINGS
Definition 3.1: f : X → Y is said to be νg-open if image of every open set in X is νg-open in Y
Theorem 3.1:
(i) Every rg-open map is νg-open.
(ii) Every rgα-open map is νg-open.
(iii) Every gr-open map is νg-open.
(iv) Every sg-open map is νg-open.
g-open Mapings
9
(v) Every gs-open map is νg-open.
(vi) Every g-open map is νg-open.
(vii) Every gα-open map is νg-open.
(viii) Every r-open map is νg-open.
(ix) Every rα-open map is νg-open.
(x) Every ν-open map is νg-open.
(xi) Every open map is νg-open.
(xii) Every α-open map is νg-open.
(xiii) Every semi-open map is νg-open.
(xiv) Every π-open map is νg-open.
Proof: (i) f is rg-open ⇒ image of every open set is rg-open ⇒ image of every open set is
νg-open[since every rg-open set is νg-open] ⇒ f is νg-open.
Similarly we can prove the remaining parts using definition 2.1 and Note 1.
Example 1: Let X = Y = {a, b, c}; τ = {φ, {a}, {b}, {a, b}, X}. and σ = {φ, {a}, Y} and let f : X
→ Y is identity map. Then f is g-open; rg-open; gr-open; sg-open; gs-open; g-open; gp-open;
rα-open; but not ν-open; r-open; open; semi-open; pre-open; pg-open; α-open; β-open.
Note 3: By note 1 and from the above theorem we have the following implication diagram.
However, we have the following converse part:
Theorem 3.2: If νgO(Y) = RO(Y) we have the following:
(i) If f is νg-open then f is r-open
(ii) If f is νg-open then f is rα-open
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International Journal of Computer, Mathematical Sciences and Applications
(iii) If f is νg-open then f is ν-open
(iv) If f is νg-open then f is rgν-open
(v) If f is νg-open then f is rg-open
(vi) If f is νg-open then f is semi-open
(vii) If f is νg-open then f is β-open
(viii) If f is νg-open then f is α-open
(ix) If f is νg-open then f is open
Theorem 3.3: If (Y, σ) is a discrete space, then f is open of all types:
Example 2: Let X = Y = {a, b, c} and τ = {φ, {b}, {a, b}, {b, c}, X}; σ = ℘(Y ) and let f : X →
Y be identity map, then f is g-open; rg-open; gr-open; sg-open; gs-open; g-open and rgα-open;
ν-open; r-open; open; pre-open; rp-open; β-open; rα-open; α-open and pg-open.
Example 3: Let X = Y = {a, b, c} and τ = ℘(X); σ = {φ, {b}, {a, b}, {b, c}, Y} and let f : X →
Y be identity map, then f is νg-open; rg-open; rgα-open but not gr-open; sg-open; gs-open; g-open;
ν-open; r-open; open; pre-open; rp-open; β-open; rα-open; α-open and pg-open.
Theorem 3.4:
(i) If f is open and g is νg-open[rg-oprn] then g  f is νg-open
(ii) If f and g are r-open then g  f is νg-open
(iii) If f is r-open and g is νg-open then g  f is νg-open
Proof: (i) Let A be open set in X ⇒ f(A) is open in Y ⇒ g(f(A)) is νg-open in Z ⇒ g 
f(A) is νg-open in Z ⇒ g  f is νg-open.
Corollary 3.2:
(i) If f is open and g is ν-open[r-open] then g  f is νg-open
(ii) If f is r-open and g is ν-open then g  f is νg-open
Theorem 3.5: If f : X → Y is νg-open, then νg(f(A))° ⊃ f(A°)
Proof: Let A ⊂ X and f : X → Y is νg-open gives f(A°) is νg-open in Y and f(A) ⊃ f(A°) which in
turn gives
νg(f(A))° ⊃ νg(f(A°))°
(1)
νg(f(A°))° = f(A°)
(2)
Since f(A°) is νg-open in Y,
Combaining (1) and (2) we have νg(f(A))° ⊃ f(A°) for every subset A of X.
Remark: Converse is not true in general, as shown by
Example 4: Let X = Y = {a, b, c}; τ = {φ, {a}, {c}, {a, c}, X} and σ = {φ, {a}, {b}, {a, b}, Y}.
f : (X, τ) → (Y, σ) be the identity map then νg(f(A))° ⊃ f(A°) for every subset A of X but f is not
νg-open. Since f({a, b}) = {a, b} is not νg-open.
g-open Mapings
11
Theorem 3.6: If f : X → Y is νg-open[rg-open] and A ⊂ X is open, then f (A) is τνg-open in Y.
Proof: Let A ⊂ X and f : X → Y is νg-open implies νg(f(A))° ⊃ f(A°) which in turn implies
νg(f(A))° ⊃ f(A), since f(A) = f(A°). But f(A) ⊃ g(f(A))°. Combaining we get f(A) = νg(f(A))°. Therefore
f (A) is τνg-open in Y.
Corollary 3.3:
(i) If f : X → Y is rg-open, then νg(f(A))° ⊃ f(A°)
(ii) If f : X → Y is ν-open[r-open], then νg(f(A))° ⊃ f(A°)
(iii) If f : X → Y is ν-open[r-open], then f (A) is τνg-open in Y if A is open[r-open] set in X.
Theorem 3.7: If νg(A)° = rg(A)° for every A ⊂ Y, then the following are equivalent:
(i) f : X → Y is νg-open map
(ii) νg(f(A))° ⊃ f(A°)
Proof: (i) ⇒ (ii) follows from theorem 3.4
(ii) ⇒ (i) Let A be any open set in X, then f(A) = f(A°) ⊂ νg(f(A))° by hypothesis. We have f(A)
⊃ νg(f(A))°. Combaining we get f(A) = νg(f(A))° = rg(f(A))°[ by given condition] which implies
f(A) is rg-open and hence νg-open. Thus f is νg-open.
Theorem 3.8: f : X → Y is νg-open iff for each subset S of Y and each open set U containing
f–1(S), there is a νg-open set V of Y such that S ⊂ V and f–1(V) ⊂ U.
Proof: Assume f is νg-open, S ⊂ Y and U an open set of X containing f–1(S), then f (X – U) is
νg-open in Y and V = Y – f (X – U) is νg-open in Y. f–1(S) ⊂ U implies S ⊂ V and f–1(V ) = X – f–1(f(X
– U)) ⊂ X – (X – U) = U.
Conversely let F be open in X, then f–1(f(Fc)) ⊂ Fc. By hypothesis, ∃ V ∈ νGO(Y ) ∋ f(Fc) ⊂ V
and f–1(V) ⊂ Fc and so F ⊂ (f–1(V))c. Hence Vc ⊂ f(F) ⊂ f[(f–1(V))c] ⊂ Vc implies f(F) ⊂ Vc, which
mplies f(F) = Vc. Thus f(F) is νg-open in Y and therefore f is νg-open.
Remark: Composition of two νg-open maps is not νg-open.
Theorem 3.9: Let X, Y, Z be spaces and every νg-open set is open[r-open] in Y, then the
composition of two νg-open maps is νg-open.
Proof: Let A be open in X ⇒ f(A) is νg-open in Y ⇒ f (A) is open in Y[by assumption] ⇒ g(f (A))
is νg-open in Z ⇒ g  f(A) is ng-open in Z ⇒ g  f is νg-open.


Theorem 3.10: If f : X → Y is g-open; g : Y → Z is νg-open[rg-open] and Y is T1  r − T1  , then
2 
2
g  f is νg-open.
Theorem 3.11: If f : X → Y is rg-open; g : Y → Z is νg-open[rg-open] and Y is r − T1 , then g
2
f
is
νg-open.



Corollary 3.4: If f : X → Y is g-open[rg-open]; g : Y → Z is ν-open[r-open] and Y is T1  r − T1  ,
2 
2
theng g  f is νg-open.
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International Journal of Computer, Mathematical Sciences and Applications
Theorem 3.12: If f : X → Y; g : Y → Z be two mappings such that g  f is νg-open[rgg-open].
Then the following are true
(i) If f is continuous[r-continuous] and surjective, then g is νg-open
(ii) If f is g-continuous, surjective and X is T1 , then g is νg-open
2
(iii) If f is g-continuous[rg-continuous], surjective and X is r − T1 , then g is νg-open.
2
Corollary 3.05: If f : X → Y; g : Y → Z be two mappings such that g  f is ν-open[r-open]. Then the
following are true
(i) If f is continuous[r-continuous] and surjective, then g is νg-open
(ii) If f is g-continuous, surjective and X is T1 , then g is νg-open
2
(iii) If f is g-continuous[rg-continuous], surjective and X is r − T1 , then g is νg-open
2
Theorem 3.13: If X is νg-regular, f : X → Y is r-open, rg-continuous, νg-open surjection and A°
= A for every νg-open set in Y, then Y is νg-regular.
Proof: Let p ∈ U ∈ νO(Y), ∃x ∈ X ∋ f(x) = p. Since X is ν-regular and f is rgg-continuous ∃V ∈
RGC(X) ∋ x ∈ V° ⊂ V ⊂ f–1(U) which implies
p ∈ f(V°) ⊂ f(V) ⊂ U
(1)
Since f is νg-open, f(V °) ⊂ U is νg-open and so f(V°)° = f(V°) and
f(V°)° = f(V)°
(2)
Combaining (1) and (2) p ∈ f(V)° ⊂ f(V) ⊂ U and f(V) is rg-open. Hence Y is νg-regular.
Corollary 3.6: If X is νg-regular, f : X → Y is r-open, rg-continuous, νg-open surjection and A°
= A for every rg-open set in Y, then Y is νg-regular.
Theorem 3.14: If f : X → Y is νg-open[rg-open] and A is open set of X, then fA : (X, τ(A)) → (Y,
σ) is νg-open.
Proof: Let F be open set in A. Then F = A ∩ E for some open set E of X and so F is open in X
which implies f (A) is νg-open in Y. But f(F) = fA(F) and therefore fA is νg-open.


Theorem 3.15: If f : X → Y is νg-open[rg-open], X is T1  rT1  and A is g-open[rgg-open] set of
2 
2
X, then fA : (X, τ(A)) → (Y, σ) is νg-open.
Corollary 3.7:
(i) If f is ν-open[r-open] and A is open set of X, then fA : (X, τ(A)) → (Y, σ) is νg-open.


(ii) If f is ν-open[r-open], X is T1  rT1  and A is g-open[rgg-open] set of X, then fA : (X, τ(A)) →
2 
2
(Y, σ) is νg-open.
g-open Mapings
13
Theorem 3.16: If fi : Xi → Yi be νg-open[rg-open] for i = 1, 2. Let f : X1 × X2 → Y1 × Y2 be
defined as follows: f(x1, x2) = (f1(x1), f2(x2)). Then f : X1 × X2 → Y1 × Y2 is νg-open.
Proof: Let U1 × U2 ⊂ X1 × X2 where Ui is open in Xi for i = 1, 2. Then f(U1 × U2) = f1(U1) × f2(U2)
a νg-open set in Y1 × Y2. Thus f(U1 × U2) is νg-open and hence f is νg-open.
Corollary 3.8: If fi : Xi → Yi be ν-open[r-open] for i = 1, 2. Let f : X1 × X2 → Y1 × Y2 be defined
as follows: f(x1, x2) = (f1(x1), f2(x2)). Then f : X1 × X2 → Y1 × Y2 is νg-open.
Theorem 3.17: Let h : X → X1 × X2 be νg-open[rg-open]. Let fi : X → Xi be defined as: h(x) =
(x1, x2) and fi(x) = xi. Then fi : X → Xi is νg-open for i = 1, 2.
Proof: Let U1 is open in X1, then U1 × X2 is open in X1 × X2, and h(U1 × X2) is νg-open in X. But
f1(U1) = h(U1 × X2), therefore f1 is νg-open. Similarly we can show that f2 is also νg-open and thus fi
: X → Xi is νg-open for i = 1, 2.
Corollary 3.09: Let h : X → X1 × X2 be ν-open[r-open]. Let fi : X → Xi be defined as: h(x) = (x1,
x2) and fi(x) = xi. Then fi : X → Xi is νg-open for i = 1, 2.
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