Volume 2008, Article ID 790964,9pages doi:10.1155/2008/790964
Research Article
Weak Forms of Continuity and Associated Properties
Mohammad S. Sarsak
Department of Mathematics, Faculty of Science, The Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan
Correspondence should be addressed to Mohammad S. Sarsak,[email protected] Received 17 July 2008; Accepted 17 September 2008
Recommended by Dalibor Froncek
We introduce slightlyp-continuous mapping and almostp-open mapping and investigate the relationships between these mappings and related types of mappings, and also study some properties of these mappings.
Copyrightq2008 Mohammad S. Sarsak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and preliminaries
A subsetAof a space X is called regular open ifA IntA, and regular closed ifX\Ais regular open, or equivalently, ifA IntA. It is well known that a subsetAof a spaceX is regular open if and only ifA IntF, whereFis closed andAis regular closed if and only ifA U, whereUis open.Ais called semi-open1 resp., preopen2, semi-preopen3, b-open4ifA⊂IntAresp.,A⊂IntA,A⊂IntA,A⊂IntA∪IntA. It is known that a set Ais semi-open if and only ifU⊂A⊂Ufor some open setU, and thatAis preopenresp., semi-preopenif and only ifAU∩D, whereUis openresp., semi-openandDis dense.
The concept of a preopen set was introduced in5, where the term locally dense was used and the concept of a semi-preopen set was introduced in6under the nameβ-open. It was pointed out in3thatAis semi-preopen if and only ifP ⊂A ⊂ P for some preopen setP.
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 andb-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-openresp., preopen, semi-preopen,b-opensets is semi-openresp., preopen, semi-preopen,b-open.Ais called semi-closedresp., preclosed, semi-preclosed,b-closedif X\Ais semi-openresp., preopen, semi-preopen,b-open. It is well known that a subsetA is regular closed if and only ifAis both closed and semi-open if and only ifAis both closed and semi-preopen.
A mappingffrom a spaceXinto a spaceYis called regular open7if it maps regular open subsets onto regular open sets, almost open8iff−1U⊂f−1UwheneverUis open inY, slightly continuous7iffU⊂fUwheneverUis open inX, semi-continuous1if the inverse image of each open set is semi-open,β-continuous6if the inverse image of each open set isβ-open, weakly continuous9if for eachx∈Xand for each open setVcontaining fxthere exists an open setUcontainingxsuch thatfU⊂V, weaklyθ-irresolute10if the inverse image of each regular closed set is semi-open, rc-continuous11if 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 indicateβ-continuous. Clearly, every semi-continuous mapping is semi-precontinuous, every rc-continuous mapping is weaklyθ-irresolute, and every weaklyθ-irresolute mapping is wrc- continuous. In7, it is shown that the properties semi-continuous and slightly continuous are independent of each other.
A spaceXis called a weakP-space13if for each countable family{Un :n∈N}of open subsets ofX,
Un
Un. Clearly,X is a weakP-space if and only if the countable union of regular closed subsets ofXis regular closedclosed.
A spaceXis called rc-Lindel ¨of14if every regular closed cover ofXhas a countable subcover, and called almost rc-Lindel ¨of15if every regular closed cover ofXhas a countable subfamily whose union is dense inX.
A subsetA of a spaceX is called anS-set inX 16if every cover of Aby regular closed subsets ofXhas a finite subcover, and called an rc-Lindel ¨of set inX resp., an almost rc-Lindel ¨of set inX 17if every cover ofAby regular closed subsets ofXadmits a countable subfamily that covers A resp., the closure of the union of whose members contains A.
Obviously, everyS-set is an rc-Lindel ¨of set and every rc-Lindel ¨of set is an almost rc-Lindel ¨of set. It is also clear that a subsetAof a weakP-spaceXis rc-Lindel ¨of inX if and only if it is almost rc-Lindel ¨of inX.
Throughout this paper,Nresp.,Q,Rdenotes the set of naturalresp., rational, real numbers. For the concepts not defined here, we refer the reader to18.
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 in19without proof. We will, however, state and prove it for its special importance in the material of our paper.
Lemma 2.1. iLetf : X→Y be a semi-continuous and almost open mapping. Thenf is weakly θ-irresolute.
ii Let f : X→Y be a semi-precontinuous and almost open mapping. Then f is wrc- continuous.
Proof. iLetUbe an open subset ofY. Sincefis almost open, thenf−1U⊂f−1U. Since f is semi-continuous, thenf−1Uis semi-open, hence there exists an open subsetV of X such thatV ⊂ f−1U ⊂ V, therefore,V ⊂ f−1U ⊂ f−1U ⊂ f−1U ⊂ V. Thusf−1Uis semi-open, andfis weaklyθ-irresolute.
iiLetUbe an open subset ofY. Sincefis almost open, thenf−1U⊂f−1U. Since fis semi-precontinuous, thenf−1Uis semi-preopen, hence there exists a preopen subsetV ofXsuch thatV ⊂f−1U⊂V, therefore,V ⊂f−1U⊂f−1U⊂f−1U⊂V. Thusf−1U is semi-preopen, andfis wrc-continuous.
Corollary 2.2see12. Letf:X→Y be a semi-continuous and almost open mapping. Thenf is wrc-continuous.
Proposition 2.3. For a mappingf :X→Y, the following are equivalent:
ifis slightly continuous;
iifU⊂fUwheneverUis semi-open inX.
Proof. Since every open set is semi-open, it suffices to show thati→ii. LetUbe a semi- open subset ofX. Then there exists an open subsetV ofXsuch thatV ⊂U⊂V. Thus byi, fU fV⊂fV⊂fU.
Proposition 2.4. Letf:X→Ybe a slightly continuous mapping. Then the following are equivalent:
ifis weaklyθ-irresolute;
iifis rc-continuous.
Proof. Since every regular closed set is semi-open, it suffices to show thati→ii. LetAbe a regular closed subset ofY. Byi,f−1Ais semi-open, butf is slightly continuous, so by Proposition 2.3,ff−1A ⊂ ff−1A ⊂ A A. Thusf−1A ⊂ f−1ff−1A ⊂ f−1A, that is,f−1Ais closed, but f−1Ais semi-open, so f−1A is regular closed. Hencef is rc-continuous.
Corollary 2.5. Letf :X→Y be a slightly continuous, semi-continuous, and almost open mapping.
Thenfis rc-continuous.
Proof. Follows fromLemma 2.1iandProposition 2.4.
Proposition 2.6. Let f : X→Y be a slightly continuous and semi-continuous mapping. Then f−1U⊂f−1Ufor every open subsetUofY.
Proof. Let Ube an open subset of Y. Since f is semi-continuous, it follows thatf−1U is semi-open, butf is slightly continuous, so it follows fromProposition 2.3thatff−1U ⊂ ff−1U⊂U. Thusf−1U⊂f−1ff−1U⊂f−1U.
The following corollary is a slight improvement ofCorollary 2.5. 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.
Thenf−1U⊂f−1Ufor every open subsetUofY.
Proposition 2.8. Letf :X→Ybe an rc-continuous mapping. IfAis rc-Lindel¨of inX, thenfAis rc-Lindel¨of inY.
Proof. Let{Uα:α∈Λ}be a cover offAby regular closed subsets ofY. Then{f−1Uα:α∈ Λ}is a cover ofAby regular closed subsets ofXasfis rc-continuous. SinceAis rc-Lindel ¨of inX, it follows that there existα1, α2, . . .∈Λsuch thatA⊂∞
i1f−1Uαi, thus it follows that fA⊂∞
i1ff−1Uαi⊂∞
i1Uαi. HencefAis rc-Lindel ¨of inY.
Corollary 2.9see19. Letf :X→Y be a slightly continuous and weaklyθ-irresolute mapping.
IfAis rc-Lindel¨of inX, thenfAis rc-Lindel¨of inY. Proof. Follows from Propositions2.4and2.8.
Proposition 2.10see20. Letf:X→Ybe a weakly continuous and almost open mapping. Then fis rc-continuous.
Corollary 2.11. Letf :X→Ybe a weakly continuous and almost open mapping. IfAis rc-Lindel¨of inX, thenfAis rc-Lindel¨of inY.
Proof. Follows from Propositions2.10and2.8.
Now, we prove the following known result using a slight modification on the previous proof.
Proposition 2.12see7. Letf :X→Ybe a slightly continuous and weaklyθ-irresolute mapping.
IfAis almost rc-Lindel¨of inX, thenfAis almost rc-Lindel¨of inY.
Proof. Let{Uα :α∈Λ}be a cover offAby regular closed subsets ofY. Sincefis slightly continuous and weakly θ-irresolute, it follows fromProposition 2.4thatf is rc-continuous and thus{f−1Uα:α∈Λ}is a cover ofAby regular closed subsets ofX. SinceAis almost rc-Lindel ¨of inX, it follows that there existα1, α2, . . . ∈Λsuch thatA ⊂ ∞
i1f−1Uαi. Now, f−1Uαi is regular closed and thus semi-open, but the arbitrary union of semi-open sets is semi-open, so∞
i1f−1Uαi is semi-open. Sincef is slightly continuous, it follows from Proposition 2.3 thatfA ⊂ ∞
i1ff−1Uαi ⊂ ∞
i1Uαi. HencefA is almost rc-Lindel ¨of inY.
Definition 2.13. A mappingffrom a spaceXinto a spaceYis said to be slightlyp-continuous iffU⊂fUwheneverUis preopen inX.
Proposition 2.14. For a mappingf:X→Y, the following are equivalent:
ifis slightlyp-continuous;
iifU⊂fUwheneverUis semi-preopen inX;
iiifU⊂fUwheneverUisb-open inX.
Proof. i→ii: LetUbe a semi-preopen subset ofX. Then there exists a preopen subsetV of Xsuch thatV ⊂U⊂V. Thus byi,fU fV⊂fV⊂fU.
ii→iii→i: follow since every preopen set isb-open and everyb-open set is semi- preopen.
Definition 2.15. A mappingf : X→Y is called brc-continuous iff−1Aisb-open for every regular closed subsetAofY.
Clearly, every weaklyθ-irresolute mapping is brc-continuous and every brc-contin- uous 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}}, andτ∗{X, φ,{a},{b, c}}. Then the identity mapping fromX, τontoX, τ∗is brc-continuous but not weaklyθ-irresoluteobserve that the regular closed subsets ofX, τ∗are the members ofτ∗, each of which is preopen and thus b-open inX, τ. However,{a}is not semi-open inX, τ.
Example 2.17. Letτube the usual topology on the set of real numbersRandτ {R, φ, A, R\ A}, whereA 0,1∩Q. Then the identity mapping fromR, τuontoR, τis wrc-continuous but not brc-continuousobserve that the regular closed subsets ofR, τare the members of τ, each of which is semi-preopen inR, τu. However,Ais notb-open inR, τu.
Proposition 2.18. Letf :X→Y be a slightlyp-continuous mapping. Then the following are equiv- alent:
ifis weaklyθ-irresolute;
iifis rc-continuous;
iiifis wrc-continuous;
ivfis brc-continuous.
Proof. ii→i→iv→iii: follow since every regular closed set is semi-open, every semi- open set isb-open and everyb-open set is semi-preopen.
iii→ii: letAbe a regular closed subset ofY. Byiii,f−1Ais semi-preopen, but f is slightlyp-continuous, so by Proposition 2.14, ff−1A ⊂ ff−1A ⊂ A A. Thus f−1A ⊂ f−1ff−1A ⊂ f−1A, that is,f−1Ais closed, butf−1A is semi-preopen, so f−1Ais regular closed. Hencefis rc-continuous.
Corollary 2.19. Let f : X→Y be a slightly p-continuous, semi-precontinuous, and almost open mapping. Thenfis rc-continuous.
Proof. Follows fromLemma 2.1iiandProposition 2.18.
Proposition 2.20. Letf:X→Y be a slightlyp-continuous and semi-precontinuous mapping. Then f−1U⊂f−1Ufor every open subsetUofY.
Proof. LetUbe an open subset ofY. Sincef is semi-precontinuous, it follows thatf−1U is semi-preopen, but f is slightly p-continuous, so it follows from Proposition 2.14 that ff−1U⊂ff−1U⊂U. Thusf−1U⊂f−1ff−1U⊂f−1U.
Observing that the closure of every semi-preopen set is regular closed, the following corollary seems a slight improvement ofCorollary 2.19.
Corollary 2.21. Let f : X→Y be a slightly p-continuous, semi-precontinuous, and almost open mapping. Thenf−1U f−1Ufor every open subsetUofY.
Obviously, every continuous mapping is both semi-continuous and slightlyp-continuous and every slightlyp-continuous mapping is slightly continuous, the converses are, however, not true as the following two examples tell.
Example 2.22. Let X {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 ofX, τ∗areX andφ. However, it is not slightly
p-continuousconsider the preopen subset{b, c}ofX, τ. We observe also that this is an example of a mapping that is both slightly continuous and semi-precontinuous but neither slightlyp-continuous nor semi-continuousobserve that{a},{a, b}are both dense and thus preopen inX, τ. However,{a}is not semi-open inX, τ. This example also shows that the converses of Propositions2.6and2.20are not true.
Example 2.23. LetX {a, b, c},τ {X, φ,{a}}, andτ∗ {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. Let f : X→Y be a slightly p-continuous and wrc-continuous mapping. If A is rc-Lindel¨of (resp., almost rc-Lindel¨of) inX, then fAis rc-Lindel¨of (resp., almost rc-Lindel¨of) inY.
Proof. We observe fromProposition 2.18that a mapping that is both slightlyp-continuous and wrc-continuous is both slightly continuous and weaklyθ-irresolutethe converse is not true asExample 2.22tells. Thus the result follows fromCorollary 2.9andProposition 2.12.
Corollary 2.25. Let f : X→Y be a slightly p-continuous, semi-precontinuous, and almost open mapping. IfAis rc-Lindel¨of (resp., almost rc-Lindel¨of) inX, thenfAis rc-Lindel¨of (resp., almost rc-Lindel¨of) inY.
Proof. Follows fromLemma 2.1iiandCorollary 2.24.
Remark 2.26. Since every dense set is preopen, one easily observes that if f is a slightlyp- continuous mapping from a spaceX onto a spaceY, thenf maps dense subsets ofX onto dense subsets ofY.
Recall that a spaceXis called submaximalresp., strongly irresolvableif every dense subset ofX is openresp., semi-open, or equivalently if, every preopen subset ofXis open resp., semi-open.
The following proposition is a direct consequence ofProposition 2.3.
Proposition 2.27. Letf :X→Y be a mapping from a strongly irresolvable spaceXinto a spaceY. Then the following are equivalent:
ifis slightlyp-continuous;
iifis slightly continuous.
3. Almostp-open mappings
Definition 3.1. A mappingf from a spaceX into a spaceY is said to be semi-regular open resp., semi-p-regular openif it maps regular open subsets onto semi-closed resp., semi- preclosedsubsets.
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}}, andτ∗{X, φ,{a},{b},{a, b}}. Then the identity mappingf fromX, τontoX, τ∗is semi-regular openobserve that the regular open subsets ofX, τare the members ofτ, each of which is semi-closed inX, τ∗; it is, however, not regular open since{a, c}is not regular open inX, τ∗.
Example 3.4. LetX {a, b, c},τ {X, φ,{a, c},{b}}, andτ∗ {X, φ,{a, b},{c}}. Then the identity mappingf fromX, τontoX, τ∗is semi-p-regular openobserve that{a, c}and {b}are preopen and thus semi-preopen inX, τ∗; it is, however, not semi-regular open since {a, c}is not semi-closed inX, τ∗.
Definition 3.5. A mapping f from a space X into a space Y is said to be almostp-open if f−1U⊂f−1UwheneverUis preopen inY.
Proposition 3.6. For a mappingf :X→Y, the following are equivalent:
ifis almost open;
iif−1U⊂f−1UwheneverUis semi-open inY.
Proof. Since every open set is semi-open, it suffices to show thati→ii. LetUbe a semi- open subset ofY. Then there exists an open subsetV ofY such thatV ⊂U⊂V. Thus byi, f−1U f−1V⊂f−1V⊂f−1U.
Proposition 3.7. For a mappingf :X→Y, the following are equivalent:
ifis almostp-open;
iif−1U⊂f−1UwheneverUis semi-preopen inY; iiif−1U⊂f−1UwheneverUisb-open inY.
Proof. i→ii: LetUbe a semi-preopen subset ofY. Then there exists a preopen subsetV of Y such thatV ⊂U⊂V. Thus byi,f−1U f−1V⊂f−1V⊂f−1U.
ii→iii→i: follow since every preopen set isb-open and everyb-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}}, andτ∗ {X, φ,{a, b},{c}}. Then the identity mappingf fromX, τontoX, τ∗is almost open and even regular openobserve that the regular open subsets ofX, τareX andφ; it is, however, not almostp-open since{b, c}is dense and thus preopen inX, τ∗but not dense inX, τ.
Proposition 3.10. For an almostp-open mappingf:X→Y, the following are equivalent:
ifis semi-p-regular open;
iisemi-regular open;
iiiregular open.
Proof. i→iii: LetAbe a regular open subset ofX. By assumption,fAis semi-preclosed, that is,Y\fAis semi-preopen. ByProposition 3.7,f−1Y\fA⊂f−1Y\fA⊂X\A X\A. Thusf−1Y\fA∩A φ and, therefore,Y \fA∩fA φ, that is, fA ⊂ IntfA, that is,fAis open, butfAis semi-preclosed, sofAis 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:
isemi-regular open;
iiregular open.
Proof. Since every regular open mapping is semi-regular open, it suffices to show that i→ii. LetAbe a regular open subset of X. By assumption,fAis semi-closed, that is, Y \fAis semi-open. ByProposition 3.6,f−1Y\fA⊂ f−1Y \fA⊂ X\A X\A.
Thusf−1Y\fA∩A φand, therefore,Y \fA∩fA φ, that is,fA⊂ IntfA, that is,fAis open, butfAis semi-closed, sofAis regular open.
Proposition 3.12see19. Letfbe an almost open and regular open mapping from a spaceXonto a spaceY. Then the following hold.
iIf for eachy∈Y,f−1yis anS-set inX, thenf−1Ais almost rc-Lindel¨of inXwhenever Ais almost rc-Lindel¨of inY.
iiIf for eachy∈Y,f−1yis rc-Lindel¨of inX, thenf−1Ais rc-Lindel¨of inXwheneverA is almost rc-Lindel¨of inY provided thatXis a weakP-space.
Corollary 3.13. Letfbe an almostp-open and semi-p-regular open mapping from a spaceXonto a spaceY. Then the following hold.
iIf for eachy∈Y,f−1yis anS-set inX, thenf−1Ais almost rc-Lindel¨of inXwhenever Ais almost rc-Lindel¨of inY.
iiIf for eachy∈Y,f−1yis rc-Lindel¨of inX, thenf−1Ais rc-Lindel¨of inXwheneverA is almost rc-Lindel¨of inY provided thatXis a weakP-space.
Proof. We observe fromProposition 3.10that a mapping that is both almostp-open and semi- p-regular open is both almost open and regular openthe converse is not true asExample 3.9 tells. Thus the result follows fromProposition 3.12.
Remark 3.14. Since every dense set is preopen, one easily observes that iff is an almostp- open mapping from a spaceXinto a spaceY, then the inverse image of a dense subset ofY is a dense subset ofX.
The following proposition is a direct consequence ofProposition 3.6.
Proposition 3.15. Letf :X→Y be a mapping from a spaceX into a strongly irresolvable spaceY. Then the following are equivalent:
ifis almostp-open;
iifis almost open.
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