Download Completely Preirresolute Functions Completely gp

International Journal of Mathematics and Computing Applications
ISSN: 0976-6790
Vol. 3, Nos. 1-2, January-December 2011, pp. 49-59
© International Science Press
COMPLETELY PREIRRESOLUTE FUNCTIONS
COMPLETELY gp-IRRESOLUTE FUNCTIONS
G.B. Navalagi
Abstract: The purpose of the present paper is to introduce the concept of completely
preirresolute, strongly p-irresolute and p-continuous in topological spaces as generalization
of strongly continuous mappings and the other two are stronger than preirresolute mappings.
1991 MSC: 54A05, 54A08, 54B05, 54B08, 54C08, 54C10, 54D10
Keywords and Phrases: Preopen sets, preclosed sets, regular open sets, regular closed
sets, completely continuous, precontinuous, almost continuous and completely
precontinous functions.
1. INTRODUCTION
In 1974, Arya and Gupta [5] introduced the notion of completely continuous functions.
In 1984, A.S. Mashhour et al [16] have defined and investigated the class of functions
called M-precontinuous functions. These functions have been further investigated
by Reilly and Vamanmurthy [25] with the name preirresolute functions.
The purpose of the present paper is to introduce the concept of completely
preirresolute, strongly p-irresolute and p-continuous in topological spaces as
generalization of strongly continuous mappings and the other two are stronger than
preirresolute mappings. In section 3, we obtain several characterizations of these
mappings, Section 4 deals with applications of these mappings. In section 5, we
introduce the new class of mappings called completely gp-irresolute mappings using
the generalized preclosed sets in between topological spaces.
2. PRELIMINARIES
Throughout the present paper, (X, τ) and (Y, σ) (or simply X and Y) denote the
topological spaces on which no separation axioms are assumed unless explicitly
stated. Let A be a subset of (X, τ). The closure of A and the interior of A are denoted
by clA and intA respectively.
A subset A is said to be regular open (resp. regular closed) if A = int clA (resp. A
= cl intA). A subset A of a space X is said to be (i) preopen [15] if A ⊂ int clA,
Department of Mathematics, K.L.E. Society’s G.H. College, Haveri-581110, Karnataka, India.
E-mail: [email protected]
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(ii) semiopen [12] if A ⊂ cl intA, (iii) α-open [22] if A ⊂ int cl intA and (iv) semipreopen
set [3] if A ⊂ clintclA. The complement of a preopen set is called preclosed [7]. The
family of all regular open (resp. regular closed, α-open, preopen, semiopen and
preclosed) sets of (X, τ) is denoted by RO(X) (resp. (,PO(X), SO(X), and PF(X)). The
family of all preopen sets of X containing a point x is denoted by PO(x). The preclosure
of A [7] is denoted by pclA (i.e. we denote it by A* [21]), is defined by the intersection
of all preclosed sets of X containing A. The preinterior of A [16] is denoted by pintA
(i.e. we denote it by A*, is the union of all preopen sets which are contained in A
[21]). A subset A of a space (X, τ) is called : (i) a generalized closed [13] (written as
g-closed) set if clA ⊆ U whenever A ⊆ U and U is open, (ii) a generalized preclosed
[23] (written as gp-closed) if A* ⊆ U whenever A ⊆ U and U is open, (iii) a generalized
semipreclosed [8] (written gsp-closed) set if (As) *⊆ U whenever A ⊆ U and U is
open. Complement of a g-closed set (resp. gp-closed set, gsp-closed set) is called
g-open [13] (resp. gp-open [23], gsp-open[8]).
The following definitions and results are useful in the sequel:
Definition 2.1: A function f : X → Y is said to be
(i) precontinuous [15] if f –1(V) ∈ PO(X) for each V ∈ σ,
(ii) completely continuous [5] if f–1(V) ∈ RO(X) for every V ∈ σ,
(iii) strongly continuous [5] if f –1(V) is clopen for every subset of V of Y,
(iv) preirresolute [25] if f –1(V) ∈ PO(X) for every V ∈ PO(Y),
(v) semiopen [6] if f(V) ∈ SO(Y) for every V ∈ τ
(vi) preclosed [7] if for each closed set of X, f(F) is preclosed set in Y,
(vii) generalized preclosed [23] (written as gp-closed) if for each closed set F of
X, f(F) is gp-closed set in Y,
(viii) pre-generalized preclosed [23] (written pre gp-closed) if for each preclosed
set F of X, f(F) is gp-closed set in Y,
(ix) generalized precontinuous [4] (written as gp-continuous) if f –1(V) is
gp-closed set in X for each closed set in Y,
(x) generalized preirresolute [4] (written as gp-irresolute) if f –1(V) if gp-closed
in X for each gp-closed set V of Y
(xi) strongly gp-continuous [4] if the inverse image of each gp-open set of Y is
open in X and
(xii) perfectly gp-continuous [4] if the inverse image of each gp-open set of Y is
clopen set in X.
Lemma 2.2 [10]: If a function f : X → Y is precontinuous and semiopen, then f is
preirresolute.
Completely Preirresolute Functions Completely gp-irresolute Functions
51
Definition 2.3 [10]: Let f : (X, τ) (Y, σ) be a function. Then a function f* :(X, τ)
→ (Y, σ) associated with f : (X, τ) → (Y, σ) is defined as follows f*(x) = f(x) for each
x ∈ X.
Definition 2.4 [10]: A function f : X → Y is called R-map if f–1(V) ∈ RO(X) for
each V ∈ RO(Y).
Definition 2.5[16]: A function f : X → Y is called M-preclosed if the image of
each preclosed set of X is preclosed set in Y.
Definition 2.6 [17]: A space X is called strongly compact if every cover of X by
preopen sets has a finite subcover.
Definition 2.7 [26]: A space X is called nearly compact if every regular open
cover has a finite subcover.
Definition 2.8 [21]: A subset A of X is said to be preregular if it is both preopen
and preclosed set.
Lemma 2.9 [23,Prop.1]: A subset A of a space X is called gp-open in X iff F ⊂
A* whenever F ⊂ A and F is closed set.
Lemma 2.10 [2]: Let X be a space, the following are hold:
(a) SO(X, τ) = SO(X, τα), SF(X, τ) = SF(X, τα)
(b) PO(X, τ) = PO(X, τα), PF(X, τ) = PF(X, τα)
(c) RO(X, τ) = RO(X, τα), RF(X, τ) = RF(X, τα)
Lemma 2.11 [23,Prop.1]: A subset A of a space X is gp-open if and only if F ⊆
A* whenever F ⊆ A and F is closed in X.
3. COMPLETELY PREIRRESOLUTE FUNCTIONS
Definition 3.1: A function f : X → Y is said to be completely preirresolute if and
only if the inverse image of each preopen set of Y is regular open in X.
Definition 3.2: A function f : X → Y is said to be strongly p-irresolute if and only
if the inverst image of each subset of Y is preregular.
Definition 3.3: A function f : X → Y is said to be p-continuous if the inverse
image of each preopen set of Y is open in X.
We, have the following implications :
(i) Every strongly continuous function
⇓
completely preirresolute
⇓
completely continuous.
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(ii) Every continuous function
⇓
strongly p-irresolute.
(iii) Every completely preirresolute
⇓
preirresolute.
(iv) Every strongly p-irresolute
⇓
Preirresolute
⇓
precontinuous.
(v) Every p-continuous
⇓
preirresolute.
In general, the converses in the above implications are not true.
Example 3.4: Every precontinuous function need not be preirresolute See
Example 3 in [25].
Example 3.5: Every preirresolute function need not be strongly p-irresolute.
Consider X = {a, b, c, d} and Y = {x, y, z}. Define,
τ = { Ø , {a}, {b, c}, {a, b, c}, X} and σ = { Ø , {x}, Y} be topologies on X and Y
respectively. Now, define a function f : (X, τ) → (Y, σ) by f(a) = x, f(b) = y and f(c) =
z = f(d). We observe that f is preirresolute but it is not strongly p-irresolute since {y,
z} ∈ P(Y), f –1({y, z}) = {b, c, d} ∈ PF(X) but ∉ PO(X).
Example 3.6: Every strongly p-irresolute function need not be strongly
continuous.
Consider the set X = {a, b, c, d} equipped with the topology τ = { Ø , {a, c}, {b,
d}, X} and that of the set Y = {p, q, r} with topology σ ={ Ø , {r}, {p, r}, {q, r}, Y}.
Then PO(X) = P(X) and PO(Y) = σ. Then f is strongly p-irresolute but it is not
strongly continuous since {p} ∈ P(Y), f –1({p}) = {d} ∉ τ.
Example 3.7: Every preirresolute map need not be completely preirresolute map
as well as completely continuous.
Consider the sets X and Y on which topologies τ and σ and map f as in Example3.5 above. Then f is preirresolute but it is not completely preirresolute since
f–1({x, y}) = {a, b} ∉ RO(X).
Completely Preirresolute Functions Completely gp-irresolute Functions
53
Example 3.8: Let X = {a, b, c, d} with topology τ = { Ø , {c}, {a, b}, {a, b, c},
X} and Y = {p, q, r} with topology σ = { Ø , {p, q}, Y}. Define f : X → Y by f(a) = p,
f(b) = f(c) = f(d) = r. Then, clearly f is precontinuous but f is not completely continuous
and (as well as f is not completely preirresolute).
Example 3.9: Every completely continuous function need not be completely
preirresolute.
Consider the set X = {a, b, c} with topology τ = { Ø , {a}, {b}, {a, b}, X} and Y
= {p, q, r} with topology σ = { Ø , {p, q}, Y}. Then, we define mapping f : X → Y by
f(a) = f(c) = p and f(b) = q. Then, f is completely continuous but it is not completely
preirresolute since f –1({p}) = {a, c} ∉ RO(X).
Now, we prove the following.
Theorem 3.10: A function f : X → Y is completely preirresolute iff the inverse
image of each preclosed set of Y is regular closed set in X.
Proof: Obvious.
We, recall the following.
Lemma 3.11 [1, lemma 2.8]: Let Y be preopn set in X. Then, H = Y ∩ U ∈
RO(Y), where U ∈ RO(X).
Now, we prove the following.
Theorem 3.12: A mapping f : X → Y is completely and Xo ∈ PO(X) then f/Xo : Xo
→ Y is completely preirresolute.
Proof: Let G ⊂ Y be a preopen set. Then by hypothesis, f –1(G) ∈ RO(X). Since
Xo ∈ PO(X) then by Lemma 3.11, we have H = Xo ∩ f –1(G) ∈ RO(Xo). Thus,
(f/Xo)–1(G) ∈ RO(Xo) which implies that f /Xo:Xo → Y is completely preirresolute.
Theorem 3.13: If f : X → Y is completely preirresolute and g : Y → Z is
preirresolute the gof : X → Z is completely preirresolute.
Proof: Obvious.
Theorem 3.14: If f : X → Y is completely preirresolute and g : Y → Z is
p-continuous function. Then gof is a completely preirresolute.
Proof: Obvious.
Theorem 3.15: If f : X → Y be an R-map and g : Y → Z be a completely
preirresolute map. Then gof : X → Z is completely preirresolute.
Proof: Let G ∈ PO(Z), g–1(G) ∈ RO(Y) since g is completely preirresolute map.
Again, as f is an R-map, f–1(g–1(G)) ∈ RO(X). Thus, for each G ∈ PO(Z), (gof)–1(G)
∈ RO(X) which shows that gof is completely preirresolute.
We, recall the following.
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Definition 3.16 [27]: A function f : X → Y is said to be almost open in the sense
of Singal and Singal (written as a.o.S) if f(U) is open in Y for each regular open set
of X.
Theorem 3.17: If f : X → Y is an a.o.S surjection and g : Y → Z is such that gof
: X → Z is completely preirresolute. Then g is p-continuous and hence preirresolute.
Proof: Let U be a preopen subset of Z. Then, (gof)–1(U) = f –1(g–1(U)) ∈ RO(X)
as gof is completely preirresolute. Again, f is a.o.S surjective, f (f –1(g–1(U))) =
(g–1(U)), which is open in Y. This shows that g is p-continuous and hence preirresolute
map.
Theorem 3.18: Let X be a space and {X | } be any family of spaces. Let f : X →
X be a map for each. Let f : X → X be a map defined by f (x) = (f(x)) for each xX,
[[f(x)] = f(x)]. If f is completely preirresolute, then so is each f.
Proof: Let P be the projection from X onto X. Then for each, f = P o f. Since f is
completely preirresolute and P being continuous and open and hence precontinuous
and semiopen. Then, by lemma 2.2, P is preirresolute. Then by Th. 3.10, each f is
completely preirresolute.
Theorem 3.19: Let f : (X, τ) → (Y, σ) be a map. Then f is completely preirresolute
if and only if f* is completely preirresolute.
Proof: Let f be completely preirresolute and U ∈ PO(Y,). Then, U PO(Y,) then
by Lemma 2.8(b). Since f is completely preirresolute, f –1(U)RO(X,). But as RO(X,)
= RO((X,), f –1(U) ∈ RO(X,). Thus, f* is completely preirresolute. Coversely, suppose
f* is completely preirresolute and U ∈ PO(Y,). Then, again by Lemma 2.8(b), U ∈
PO (Y,). By hypothesis, (f*)–1(U) RO(X,) = RO(X,). This shows that f is completely
preirresolute.
4. APPLICATIONS OF COMPLETELY PREIRRESOLUTE FUNCTIONS
We, define the following.
Definition 4.1: A cover U = {U|} of subsets of X is called a p-cover if U is
preopen for each.
Definition 4.2: A space X is called p-compact if each p-cover of it has a finite
subcover.
Now, we prove the following.
Theorem 4.3: Every completely preirresolute surjective image of nearly compact
space is p-compact.
Proof: Let f : X → Y be a completely preirresolute surjective map from a nearly
compact space X ont a space Y. Let U = {G|} be a p-cover of Y. Then {f–1(G) |} be a
Completely Preirresolute Functions Completely gp-irresolute Functions
55
regular open cover of X. As X is nearly compact, there exists a finite subcover of
{f–1(G) |} for X. Then it follows that is finite subcover of {G|} for Y. Hence Y is
p-compact.
We, recall the following.
Definition 4.4 [20]: A space X is called mildly p-normal if for each pair of
disjoint regular closed subsets E and F of X, there exist disjoint preopen sets U and
V such that E ⊂ U and F ⊂ V.
Definition 4.5 [19]: A space X is called strongly normal if each pair of disjoint
preclosed sets of X are separated by preopen sets.
We, recall the following.
Lemma 4.5 [20, lemma 3.10]: A map f : X → Y is M-preclosed iff for each
subset B of Y and each preopen set V containing B such that f –1(V) ⊂ U.
We, now prove the following.
Theorem 4.6: If f : X → Y be completely preirresolute M-preclosed mapping
from a mildly p-normal space X onto a space Y, then Y is strongly normal.
Proof: Let G and H be two disjoint preclosed sets of Y. Since f is completely
preirresolute, f –1(G) and f –1(H) are disjoint regular closed sets in X. Since X is
mildly p-normal space, there exist disjoint preopen sets U and V such that f–1(G) ⊂ U
and f –1(H) ⊂ V. Again, f is M-preclosed map, by lemma 4.5, there exist preopen sets
A and B such that GA and HB with, f –1(A) U and f –1(B) V. As U and V are disjoint, AB
= . Therefore, Y is strongly normal.
Now, recall the following.
Definition 4.7: A space X is called almost p-regular [18] (resp. strongly regular
[19]) if for each regular closed (resp.preclosed) set and xF, there exist disjoint preopen
sets U and V such that xU and FV.
Now, we have the following result.
Theorem 4.8: If f is a completely preirresolute M-preclosed function from an
almost p-regular space X onto a space Y, then Y is strongly regular.
We, recall the following.
Definition 4.9 [24]: A space X is called preconnected if X = AB where APO(X),
BPO(X), A, B implies AB.
In [24], it is observed that every preconnected space is connected.
We, now prove the following.
Theorem 4.10: If f : X → Y be a completely preirresolute surjection map and X
is preconnected then Y is preconnected.
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Proof: Suppose Y is not preconnected. Then Y is the union of two non-empty
disjoint preopen sets U and V. Then f–1(U) and f–1(V) are non-empty regular open
sets and hence preopen whose union is X. Hence X is not preconnected. This
contradicts that Y is not preconnected assumed. Hence Y is preconnected.
We, recall the following.
Definition 4.11 [11]: A space X is said to be pre-T2 if every pair of distinct
points of X are separated by disjoint preopen sets.
From the above definition, it is clear that if X is pre-T2 space then for each pair
of distinct points x and y of X, there exists a preopen set U containing x such that y
U*.
Now, we define the following.
Definition 4.12: The graph G(f) is said to be strongly preclosed if for each (x, y)
G(f), there exist UPO(x) and VPO(y) such that [U × V*] G(f) = .
It is clear from the above definition that G(f) is strongly preclosed iff for each (x,
y) G(f), there exist UPO(x) and VPO(y) such that f(U)V* = .
Now, we prove the following.
Theorem 4.13: If f : X → Y is completely preirresolute and Y is pre-T2 then G(f)
is strongly preclosed.
Proof: Let x ∈ X and y ∈ Y be such that y ∉ f(x), that is, (x, y) ∈ G(f). Since Y is
pre-T2, there exists a preopen set V containing y such that f(x) V*. Thus, f–1(V*) is
regular closed set of X as f is completely preirresolute and x ∉ f –1(V*). Then, x ∈ X
– f –1(V*) = U, say, which is preopen set in X. Further, U ∩ f –1(V*) = Ø or f(U) ∩ V*
= Ø . This shows that G(f) is strongly preclosed.
5. COMPLETELY gp-IRRESOLUTE FUNCTIONS
In this section, using generalized preopen (gp-open) sets, we define the new class of
mappings called completely gp-irresolute, strongly gp-irresolute and gp*-continuous
mappings in topology.
Definition 5.1: A function f : X → Y is said to be completely gp-irresolute if the
inverse image of each gp-open set is regular open in X.
Definition 5.2: A subset A of X is said to be pre gp-regular if it is both gp-open
and gp-closed set.
Definition 5.3: A function f : X → Y is said to be strongly gp-irresolute if the
inverse image of each subset of Y is pre gp-regular.
Definition 5.4: A function f : X → Y is said to be gp*-continuous if the inverse
image of each gp-open set of Y is preopen in X.
Completely Preirresolute Functions Completely gp-irresolute Functions
57
Theorem 5.5: If a function f : X → Y is completely gp-irresolute, then f is
completely continuous.
Proof is obvious.
Theorem 5.6: Every gp*-continuous function is p-continuous.
Theorem 5.7: If a function f : X → Y is strongly continuous, then f is strongly
gp-irresolute.
Proof: Let V be a subset of Y then f –1(V) is clopen set in X since f is strongly
continuous. Since every closed set gp-closed and open set is gp-open set and hence
every clopen set is pre gp-regular. This shows that f is strongly gp-irresolute function.
We, recall the following.
Lemma 5.8 [23, Prop. 3]: If f : X → Y is continuous pre gp-closed and A is
gp-closed set in X, then f(A) is gp-closed set in Y.
Lemma 5.9 [23, Prop.4]: If f : X → Y is an open preirresolute bijection and B is
gp-closed set in Y, then f –1(B) is gp-closed set in X.
We, define the following.
Definition 5.10: A space X is called generalized pre-normal (i.e., gp-normal) if
for any pair of disjoint closed sets A, B of X, there exist disjoint gp-open sets U, V
such that A ⊂ U and B ⊂ V.
Every pre-normal space is gp-normal.
We, define the following.
Definition 5.11: Generalized pre-closure of a subset A of space X is the
intersection of all gp-closed sets containing A and is denoted by gpclA.
Since every preclosed set is gp-closed and hence gpclA ⊂ pclA ⊂ clA.
Definition 5.12: Generalized semipre-closure of a subset A of space X is the
intersection of all gsp-closed sets containing A and is denoted by gpclA.
Since every preclosed set is gp-closed and every gp-closed set is gsp-closed set.
Hence gspclA ⊂ gpclA ⊂ pclA ⊂ clA.
We, give the characterizations of gp-normal spaces in the following.
Theorem 5.13: For a space X, the following statements are equivalent:
(a) X is gp-normal.
(b) For any pair of disjoint cloed sets A, B of X, there exist disjoint gsp-open
sets U, V such that A ⊂ U and B ⊂ V.
(c) For any closed set A and any open set V containing A, there a exists
gsp-open set U such that A ⊂ U ⊂ gspclA ⊂ V.
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(d) For any closed set A and any open set V containing A, there a exists gp-open
set U such that A ⊂ U ⊂ gpclA ⊂ V.
Proof (a)
(b): Proof is obvious since every gp-open set is gsp-open set.
(b) ⇒ (c): Let A be any closed set and V be an open set containing A. Since A and
X – V are disjoint closed sets of X, there exist gsp-open sets U and W of X such that
A ⊂ U and X – V ⊂ W and U ∩ W = Ø by (b). Since W is gsp-open, then by definition,
we have X – V ⊂ spintW. Since U ∩ spintW = Ø and thus we have gspclU ∩ spintW
= Ø and hence gspclU ⊂ X – spintW ⊂ V. Therefore, we obtain that A ⊂ U ⊂ gspclU
⊂ V.
(c) ⇒ (d): Proof is easy as gspclA ⊂ gpclA for A ⊂ X.
(d) ⇒ (a): Let A and B be any two disjoint closed sets of X. Since X – B is an
open set containing A, there exists a gp-open set G such that A ⊂ G ⊂ gpclG ⊂ X–B.
Then, again by definition of gp-open set, we have A ⊂ A*. Now, put U = G* and V =
X–gpclG. Then, U and V are disjoint gp-open sets such that A ⊂ U and B ⊂ V. Thus,
X is gp-normal.
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