# Weak Forms of Continuity and  Associated Properties

Authors: Mohammad S. Sarsak
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

We introduce slightlyp-continuous mapping and almost p-open mapping and investigate the relationships between these mappings and related types of mappings, and also study some properties of these mappings.


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## Body

## 1. Introduction and Preliminaries

A
subsetA of a space X is called regular open if A=IntA¯, and
regular closed if  X∖A is regular open, or equivalently, if A=IntA¯. It is
well known that a subset A of a space X is regular open if and only if A=IntF, where
F is closed and A is regular closed if and only if A=U¯, where
U is open. A is called semi-open [1] (resp., preopen [2], semi-preopen [3],
b-open [4]) if  A⊂IntA¯ (resp., A⊂IntA¯, A⊂IntA¯¯, A⊂IntA¯∪IntA¯). It
is known that a set A is semi-open if and only if U⊂A⊂U¯  for some open set U, and that A is preopen
(resp., semi-preopen) if and only if A=U∩D, where
U is open (resp., semi-open) and D is dense. The concept of a preopen set was
introduced in [5], where the term locally dense was used and the concept of a semi-preopen set was
introduced in [6] under the name β-open.
It was pointed out in [3] that A is semi-preopen if and only if P⊂A⊂P¯  for some preopen set P. Clearly, every regular
closed set is semi-open, every open set is both semi-open and preopen, semi-open
sets as well as preopen sets are b-open and b-open sets are semi-preopen. It is
also known that the closure of every semi-preopen set is regular closed and
that the arbitrary union of semi-open (resp., preopen, semi-preopen, b-open) sets
is semi-open (resp., preopen, semi-preopen, b-open). A is called semi-closed
(resp., preclosed, semi-preclosed, b-closed) if  X∖A is semi-open (resp., preopen, semi-preopen,
b-open). It is well known that a subset A is regular closed if and only if A
is both closed and semi-open if and only if A is both closed and semi-preopen.A
mappingf from a space X into a space Y is called regular open [7] if it maps
regular open subsets onto regular open sets, almost open [8] if f−1(U¯)⊂f−1(U)¯ whenever U is open in Y, slightly continuous
[7] if  f(U¯)⊂f(U)¯ whenever U is open in X, semi-continuous [1]
if the inverse image of each open set is semi-open, β-continuous
[6] if the inverse image of each open set is β-open,
weakly continuous [9] if for each  x∈X and for each open set V containing  f(x) there
exists an open set U containing x such that f(U)⊂V¯,
weakly θ-irresolute [10] if the inverse
image of each regular closed set is semi-open, rc-continuous [11] if the inverse
image of each regular closed set is regular closed, and wrc-continuous [12] if
the inverse image of each regular closed set is semi-preopen. We will use the
term semi-precontinuous to indicate β-continuous.
Clearly, every semi-continuous mapping is semi-precontinuous, every
rc-continuous mapping is weakly θ-irresolute, and every weakly θ-irresolute mapping is
wrc-continuous. In [7], it is shown that the properties semi-continuous and
slightly continuous are independent of each other.A
spaceX is called a weak P-space [13] if for each countable family  {Un:n∈N} of open subsets of X, ⋃Un¯=⋃U¯n.
Clearly, X is a weak P-space if and only if the countable union of regular
closed subsets of X is regular closed (closed).A
spaceX is called rc-Lindelöf [14] if every regular closed cover of X has a
countable subcover, and called almost rc-Lindelöf [15] if every regular closed
cover of X has a countable subfamily whose union is dense in X.A
subsetA of a space X is called an S-set in X [16] if every cover of A by
regular closed subsets of X has a finite subcover, and called an rc-Lindelöf
set in X (resp., an almost rc-Lindelöf set in X) [17] if every cover of A by
regular closed subsets of X admits a countable subfamily that covers A (resp.,
the closure of the union of whose members contains A). Obviously, every S-set
is an rc-Lindelöf set and every rc-Lindelöf set is an almost rc-Lindelöf set.
It is also clear that a subset A of a weak P-space X is rc-Lindelöf in X if and
only if it is almost rc-Lindelöf in X.Throughout
this paper,N (resp., Q, R) denotes the set of natural (resp., rational, real)
numbers. For the concepts not defined here, we refer the reader to [18].


## 2. Slightlyp-continuous Mappings

This
section is mainly devoted to study several properties of slightlyp-continuous
mappings. Now, we begin with the following lemma which was pointed out in [19]
without proof. We will, however, state and prove it for its special importance
in the material of our paper.Lemma 2.1.
(i) Letf:X→Y be a semi-continuous and almost open mapping. Then f is weakly θ-irresolute.
(ii) Letf:X→Y be a semi-precontinuous and almost open mapping. Then f is wrc-continuous.Proof..
(i) LetU be an open subset of Y. Since f is almost open, then f−1(U¯)⊂f−1(U)¯. Since
f is semi-continuous, then f−1(U)  is semi-open, hence there exists an open subset
V of X such that V⊂f−1(U)⊂V¯,
therefore, V⊂f−1(U)⊂f−1(U¯)⊂f−1(U)¯⊂V¯. Thus f−1(U¯)  is semi-open, and f is weakly θ-irresolute.
(ii) LetU be an open subset of Y. Since f is
almost open, then f−1(U¯)⊂f−1(U)¯. Since
f is semi-precontinuous, then  f−1(U) is semi-preopen, hence there exists a preopen
subset V of X such that V⊂f−1(U)⊂V¯,
therefore, V⊂f−1(U)⊂f−1(U¯)⊂f−1(U)¯⊂V¯. Thus  f−1(U¯) is semi-preopen, and f is wrc-continuous.Corollary 2.2 (see [12]).
Letf:X→Y be a semi-continuous and almost open mapping. Then f is wrc-continuous.Proposition 2.3.
For a mappingf:X→Y, the following are equivalent:
(i)
f is slightly continuous;(ii)
f(U¯)⊂f(U)¯ whenever U is semi-open in X.Proof..
Since every open set is semi-open, it suffices to show that  (i)→(ii). Let U be a semi-open subset of X. Then there
exists an open subset V of X such that  V⊂U⊂V¯. Thus by (i), f(U¯)=f(V¯)⊂f(V)¯⊂f(U)¯.Proposition 2.4.
Letf:X→Y be a slightly continuous mapping. Then the following are equivalent:
(i)
f is weakly θ-irresolute;(ii)
f is rc-continuous.Proof..
Since every regular closed set is semi-open, it suffices to show that  (i)→(ii). Let A be a regular closed subset of Y. By (i),
f−1(A) is semi-open, but f is slightly continuous, so
by Proposition 2.3,  f(f−1(A)¯)⊂f(f−1(A))¯⊂A¯=A. Thus f−1(A)¯⊂f−1f(f−1(A)¯)⊂f−1(A), that
is,  f−1(A) is closed, but  f−1(A) is semi-open, so  f−1(A) is regular closed. Hence f is rc-continuous.Corollary 2.5.
Letf:X→Y be a slightly continuous, semi-continuous, and
almost open mapping. Then f is rc-continuous.Proof..
Follows from Lemma2.1(i) and Proposition 2.4.Proposition 2.6.
Letf:X→Y be a slightly continuous and semi-continuous
mapping. Then f−1(U)¯⊂f−1(U¯) for every open subset U of Y.Proof..
LetU be an open subset of Y. Since f is semi-continuous, it follows that  f−1(U) is semi-open, but f is slightly continuous, so
it follows from Proposition 2.3 that f(f−1(U)¯)⊂f(f−1(U))¯⊂U¯. Thus f−1(U)¯⊂f−1f(f−1(U)¯)⊂f−1(U¯).The following corollary is a slight improvement of Corollary2.5. This is because the closure of every semi-open set is regular closed.Corollary 2.7.
Letf:X→Y be a slightly continuous, semi-continuous, and
almost open mapping. Then f−1(U)¯⊂f−1(U¯) for every open subset U of Y.Proposition 2.8.
Letf:X→Y be an rc-continuous mapping. If A is
rc-Lindelöf in X, then f(A) is rc-Lindelöf in Y.Proof..
Let{Uα:α∈Λ} be a cover of f(A) by regular closed subsets
of Y. Then  {f−1(Uα):α∈Λ} is a cover of A by regular closed subsets of X
(as f is rc-continuous). Since A is rc-Lindelöf in X, it follows that there
exist  α1,α2,…∈Λ such that A⊂⋃i=1∞f−1(Uαi), thus
it follows that f(A)⊂⋃i=1∞f(f−1(Uαi))⊂⋃i=1∞Uαi. Hence
f(A) is rc-Lindelöf in Y.Corollary 2.9 (see [19]).
Letf:X→Y be a slightly continuous and weakly θ-irresolute mapping. If A is
rc-Lindelöf in X, then f(A) is rc-Lindelöf in Y.Proof..
Follows from Propositions2.4 and 2.8.Proposition 2.10 (see [20]).
Letf:X→Y be a weakly continuous and almost open
mapping. Then f is rc-continuous.Corollary 2.11.
Letf:X→Y be a weakly continuous and almost open
mapping. If A is rc-Lindelöf in X, then f(A) is rc-Lindelöf in Y.Proof..
Follows from Propositions2.10 and 2.8.Now, we prove the following known result using a slight modification on the previous proof.Proposition 2.12 (see [7]).
Letf:X→Y be a slightly continuous and weakly θ-irresolute mapping. If A is
almost rc-Lindelöf in X, then f(A) is almost rc-Lindelöf in Y.Proof..
Let{Uα:α∈Λ} be a cover of
f(A) by regular closed subsets of Y. Since f is slightly continuous and
weakly θ-irresolute, it follows from Proposition 2.4 that f is rc-continuous and thus  {f−1(Uα):α∈Λ} is a cover of A by regular closed subsets of
X. Since A is almost rc-Lindelöf in X, it follows that there exist  α1,α2,…∈Λ such that A⊂⋃i=1∞f−1(Uαi)¯. Now,  f−1(Uαi) is regular closed and thus semi-open, but the
arbitrary union of semi-open sets is semi-open, so  ⋃i=1∞f−1(Uαi) is semi-open. Since f is slightly continuous,
it follows from Proposition 2.3 that f(A)⊂⋃i=1∞f(f−1(Uαi))¯⊂⋃i=1∞Uαi¯. Hence
f(A) is almost rc-Lindelöf in Y.Definition 2.13.
A mappingf from a space X into a space Y is said to be slightly p-continuous if  f(U¯)⊂f(U)¯ whenever U is preopen in X.Proposition 2.14.
For a mappingf:X→Y, the
following are equivalent:
(i)
f is slightly p-continuous;(ii)
f(U¯)⊂f(U)¯ whenever U is semi-preopen in X;(iii)
f(U¯)⊂f(U)¯ whenever U is b-open in X.Proof..
(i)→(ii): Let U be a semi-preopen subset of X. Then
there exists a preopen subset V of X such that  V⊂U⊂V¯. Thus by (i), f(U¯)=f(V¯)⊂f(V)¯⊂f(U)¯.
(ii)→(iii)→(i): follow since every preopen set is b-open and
every b-open set is semi-preopen.Definition 2.15.
A mappingf:X→Y is called brc-continuous if  f−1(A) is b-open for every regular closed subset A of
Y.Clearly,
every weaklyθ-irresolute mapping is
brc-continuous and every brc-continuous mapping is wrc-continuous; the
converses are, however, not true as the following two examples tell.Example 2.16.
LetX={a,b,c}, τ={X,ϕ,{a,b}},  and τ*={X,ϕ,{a},{b,c}}. Then
the identity mapping from  (X,τ) onto  (X,τ*) is brc-continuous but not weakly θ-irresolute (observe that the
regular closed subsets of  (X,τ*) are the members of τ*, each
of which is preopen and thus b-open in (X,τ).
However,  {a} is not semi-open in (X,τ).Example 2.17.
Letτu be the usual topology on the set of real
numbers R and τ={R,ϕ,A,R∖A}, where
A=[0,1)∩Q. Then
the identity mapping from  (R,τu) onto  (R,τ) is wrc-continuous but not brc-continuous
(observe that the regular closed subsets of  (R,τ) are the members of τ, each
of which is semi-preopen in (R,τu).
However, A is not b-open in (R,τu).Proposition 2.18.
Letf:X→Y be a slightly p-continuous mapping. Then the
following are equivalent:(i)
f is weakly θ-irresolute;(ii)
f is rc-continuous;(iii)
f is wrc-continuous;(iv)
f is brc-continuous.Proof..
(ii)→(i)→(iv)→(iii): follow
since every regular closed set is semi-open, every semi-open set is b-open and
every b-open set is semi-preopen.
(iii)→(ii): let A
be a regular closed subset of Y. By (iii),  f−1(A) is semi-preopen, but f is slightly
p-continuous, so by Proposition 2.14,  f(f−1(A)¯)⊂f(f−1(A))¯⊂A¯=A. Thus f−1(A)¯⊂f−1f(f−1(A)¯)⊂f−1(A), that
is,  f−1(A) is closed, but  f−1(A) is semi-preopen, so  f−1(A) is regular closed. Hence f is rc-continuous.Corollary 2.19.
Letf:X→Y be a slightly p-continuous, semi-precontinuous,
and almost open mapping. Then f is rc-continuous.Proof..
Follows from Lemma2.1(ii) and Proposition 2.18.Proposition 2.20.
Letf:X→Y be a slightly p-continuous and
semi-precontinuous mapping. Then f−1(U)¯⊂f−1(U¯) for every open subset U of Y.Proof..
LetU be an open subset of Y. Since f is semi-precontinuous, it follows that  f−1(U) is semi-preopen, but f is slightly
p-continuous, so it follows from Proposition 2.14 that f(f−1(U)¯)⊂f(f−1(U))¯⊂U¯. Thus f−1(U)¯⊂f−1f(f−1(U)¯)⊂f−1(U¯).Observing
that the closure of every semi-preopen set is regular closed, the following
corollary seems a slight improvement of Corollary2.19.Corollary 2.21.
Letf:X→Y be a slightly p-continuous, semi-precontinuous,
and almost open mapping. Then f−1(U)¯=f−1(U¯) for every open subset U of Y.
Obviously,
every continuous mapping is both semi-continuous and slightlyp-continuous and
every slightly p-continuous mapping is slightly continuous, the converses are,
however, not true as the following two examples tell.Example 2.22.
LetX={a,b,c}, τ={X,ϕ,{a,b}}, and τ*={X,ϕ,{a},{a,b}}. Then
the identity mapping from  (X,τ) onto (X,τ*)  is slightly continuous and weakly θ-irresolute (observe that the
regular closed subsets of  (X,τ*) are X and ϕ).
However, it is not slightly p-continuous (consider the preopen subset  {b,c} of (X,τ)). We
observe also that this is an example of a mapping that is both slightly
continuous and semi-precontinuous but neither slightly p-continuous nor semi-continuous
(observe that  {a}, {a,b} are both dense and thus preopen in (X,τ).
However,  {a} is not semi-open in (X,τ)). This
example also shows that the converses of Propositions 2.6 and 2.20 are not true.Example 2.23.
LetX={a,b,c}, τ={X,ϕ,{a}}, and τ*={X,ϕ,{a,b}}. Then
the identity mapping from (X,τ) onto (X,τ*) is
slightly p-continuous (observe that the nonempty preopen subsets of  (X,τ) are the supersets of {a}); it
is, moreover, semi-continuous and almost open. However, it is not continuous.Corollary 2.24.
Letf:X→Y be a slightly p-continuous and wrc-continuous
mapping. If A is rc-Lindelöf (resp., almost rc-Lindelöf) in X, then f(A) is
rc-Lindelöf (resp., almost rc-Lindelöf) in Y.Proof..
We observe from Proposition2.18 that a mapping that is both slightly
p-continuous and wrc-continuous is both slightly continuous and weakly θ-irresolute (the converse is
not true as Example 2.22 tells). Thus the result follows from Corollary 2.9 and
Proposition 2.12.Corollary 2.25.
Letf:X→Y be a slightly p-continuous, semi-precontinuous,
and almost open mapping. If A is rc-Lindelöf (resp., almost rc-Lindelöf) in X,
then f(A) is rc-Lindelöf (resp., almost rc-Lindelöf) in Y.Proof..
Follows from Lemma2.1(ii) and Corollary 2.24.Remark 2.26.
Since every dense set is preopen, one easily observes that iff is a
slightly p-continuous mapping from a space X onto a space Y, then f maps dense
subsets of X onto dense subsets of Y.
Recall
that a spaceX is called submaximal (resp., strongly irresolvable) if every
dense subset of X is open (resp., semi-open), or equivalently if, every preopen
subset of X is open (resp., semi-open).The following proposition is a direct consequence of Proposition2.3.Proposition 2.27.
Letf:X→Y be a mapping from a strongly irresolvable
space X into a space Y. Then the following are equivalent:
(i)
f is slightly p-continuous;(ii)
f is slightly continuous.


## 3. Almostp-open Mappings

Definition 3.1.
A mappingf from a space X into a space Y is said to be semi-regular open (resp., semi-p-regular open) if it maps regular open subsets onto semi-closed
(resp., semi-preclosed) subsets.Remark 3.2.
Since every regular open set is semi-closed and every semi-closed
set is semi-preclosed, it is obvious that every regular open mapping is semi-regular open and every semi-regular open mapping is semi-p-regular open. The converses
are, however, not true as the following examples show.Example 3.3.
LetX={a,b,c}, τ={X,ϕ,{a,c},{b}}, and τ*={X,ϕ,{a},{b},{a,b}}. Then
the identity mapping f from  (X,τ) onto  (X,τ*) is semi-regular open (observe that the regular
open subsets of  (X,τ) are the members of τ, each
of which is semi-closed in (X,τ*)); it
is, however, not regular open since  {a,c} is not regular open in (X,τ*).Example 3.4.
LetX={a,b,c}, τ={X,ϕ,{a,c},{b}}, and τ*={X,ϕ,{a,b},{c}}. Then
the identity mapping f from  (X,τ) onto  (X,τ*) is semi-p-regular open (observe that  {a,c} and  {b} are preopen and thus semi-preopen in (X,τ*)); it
is, however, not semi-regular open since  {a,c} is not semi-closed in (X,τ*).Definition 3.5.
A mappingf from a space X into a space Y is
said to be almost p-open if  f−1(U¯)⊂f−1(U)¯ whenever U is preopen in Y.Proposition 3.6.
For a mappingf:X→Y, the
following are equivalent:
(i)
f is almost open;(ii)
f−1(U¯)⊂f−1(U)¯ whenever U is semi-open in Y.Proof..
Since
every open set is semi-open, it suffices to show that  (i)→(ii). Let U be a semi-open subset of Y. Then there
exists an open subset V of Y such that  V⊂U⊂V¯. Thus by (i), f−1(U¯)=f−1(V¯)⊂f−1(V)¯⊂f−1(U)¯.Proposition 3.7.
For a mappingf:X→Y, the
following are equivalent:
(i)
f is almost p-open;(ii)
f−1(U¯)⊂f−1(U)¯ whenever U is semi-preopen in Y;(iii)
f−1(U¯)⊂f−1(U)¯ whenever U is b-open in Y.Proof..
(i)→(ii): Let U be a semi-preopen subset of Y. Then
there exists a preopen subset V of Y such that  V⊂U⊂V¯. Thus by (i), f−1(U¯)=f−1(V¯)⊂f−1(V)¯⊂f−1(U)¯.
(ii)→(iii)→(i): follow since every preopen set is b-open and
every b-open set is semi-preopen.Remark 3.8.
Since every open set is preopen, it is obvious that every
almostp-open mapping is almost open. However, the converse is not true as the
following example tells.Example 3.9.
LetX={a,b,c}, τ={X,ϕ,{a}}, and τ*={X,ϕ,{a,b},{c}}. Then
the identity mapping f from  (X,τ) onto  (X,τ*) is almost open and even regular open (observe
that the regular open subsets of  (X,τ) are X and ϕ); it
is, however, not almost p-open since  {b,c} is dense and thus preopen in (X,τ*)  but not dense in (X,τ).Proposition 3.10.
For an almostp-open mapping f:X→Y, the
following are equivalent:
(i)
f is semi-p-regular open;(ii)
semi-regular open;(iii)
regular open.Proof..
(i)→(iii): Let A be a regular open subset of X. By
assumption,  f(A) is semi-preclosed, that is,  Y∖f(A) is semi-preopen. By Proposition 3.7, f−1(Y∖f(A)¯)⊂f−1(Y∖f(A))¯⊂X∖A¯=X∖A. Thus  f−1(Y∖f(A)¯)∩A=ϕ and, therefore, (Y∖f(A)¯)∩f(A)=ϕ, that
is, f(A)⊂Intf(A), that
is,  f(A) is open, but  f(A) is semi-preclosed, so  f(A) is regular open.
(iii)→(ii)→(i): follow since every regular open mapping is
semi-regular open and every semi-regular open mapping is semi-p-regular open.Proposition 3.11.
For an almost open mappingf:X→Y, the
following are equivalent:
(i)
semi-regular open;(ii)
regular open.Proof..
Since every regular open mapping is semi-regular open, it suffices to
show that  (i)→(ii). Let A be a regular open subset of X. By
assumption,  f(A) is semi-closed, that is,  Y∖f(A) is semi-open. By Proposition 3.6, f−1(Y∖f(A)¯)⊂f−1(Y∖f(A))¯⊂X∖A¯=X∖A. Thus  f−1(Y∖f(A)¯)∩A=ϕ and, therefore, (Y∖f(A)¯)∩f(A)=ϕ, that
is, f(A)⊂Intf(A), that
is,  f(A) is open, but  f(A) is semi-closed, so  f(A) is regular open.Proposition 3.12 (see [19]).
Letf be an almost open and regular open mapping from a
space X onto a space Y. Then the following hold.
(i)
If for eachy∈Y, f−1(y) is an S-set in X, then f−1(A) is almost rc-Lindelöf in X whenever A is
almost rc-Lindelöf in Y.(ii)
If for eachy∈Y, f−1(y) is rc-Lindelöf in X, then f−1(A) is rc-Lindelöf in X whenever A is almost
rc-Lindelöf in Y provided that X is a weak P-space.Corollary 3.13.
Letf be an almost p-open and
semi-p-regular open mapping from a space X onto a space Y. Then the following
hold.
(i)
If for eachy∈Y, f−1(y) is an S-set in X, then f−1(A) is almost rc-Lindelöf in X whenever A is
almost rc-Lindelöf in Y.(ii)
If for eachy∈Y, f−1(y) is rc-Lindelöf in X, then f−1(A) is rc-Lindelöf in X whenever A is almost rc-Lindelöf in Y provided that X is a weak P-space.Proof..
We
observe from Proposition3.10 that a mapping that is both almost p-open and
semi-p-regular open is both almost open and regular open (the converse is not true as Example 3.9 tells). Thus the result follows from Proposition 3.12.Remark 3.14.
Since every dense set is preopen, one easily observes that iff is an
almost p-open mapping from a space X into a space Y, then the inverse image of
a dense subset of Y is a dense subset of X.The following proposition is a direct consequence of Proposition3.6.Proposition 3.15.
Letf:X→Y be a mapping from a space X into a strongly
irresolvable space Y. Then the following are equivalent:
(i)
f is almost p-open;(ii)
f is almost open.



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