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International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 790964, 9 pages
http://dx.doi.org/10.1155/2008/790964
Research Article

Weak Forms of Continuity and Associated Properties

Department of Mathematics, Faculty of Science, The Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan

Received 17 July 2008; Accepted 17 September 2008

Academic Editor: Dalibor Froncek

Copyright © 2008 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.

Abstract

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

1. Introduction and Preliminaries

A subset of a space is called regular open if , and regular closed if is regular open, or equivalently, if . It is well known that a subset of a space is regular open if and only if , where is closed and is regular closed if and only if , where is open. is called semi-open [1] (resp., preopen [2], semi-preopen [3], -open [4]) if (resp., , , ). It is known that a set is semi-open if and only if for some open set , and that is preopen (resp., semi-preopen) if and only if , where is open (resp., semi-open) and 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 -open. It was pointed out in [3] that is semi-preopen if and only if for some preopen set . 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 -open and -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, -open) sets is semi-open (resp., preopen, semi-preopen, -open). is called semi-closed (resp., preclosed, semi-preclosed, -closed) if is semi-open (resp., preopen, semi-preopen, -open). It is well known that a subset is regular closed if and only if is both closed and semi-open if and only if is both closed and semi-preopen.

A mapping from a space into a space is called regular open [7] if it maps regular open subsets onto regular open sets, almost open [8] if whenever is open in , slightly continuous [7] if whenever is open in , semi-continuous [1] if the inverse image of each open set is semi-open, -continuous [6] if the inverse image of each open set is -open, weakly continuous [9] if for each and for each open set containing there exists an open set containing such that , weakly -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 -continuous. Clearly, every semi-continuous mapping is semi-precontinuous, every rc-continuous mapping is weakly -irresolute, and every weakly -irresolute mapping is wrc-continuous. In [7], it is shown that the properties semi-continuous and slightly continuous are independent of each other.

A space is called a weak -space [13] if for each countable family of open subsets of , . Clearly, is a weak -space if and only if the countable union of regular closed subsets of is regular closed (closed).

A space is called rc-Lindelöf [14] if every regular closed cover of has a countable subcover, and called almost rc-Lindelöf [15] if every regular closed cover of has a countable subfamily whose union is dense in .

A subset of a space is called an -set in [16] if every cover of by regular closed subsets of has a finite subcover, and called an rc-Lindelöf set in (resp., an almost rc-Lindelöf set in ) [17] if every cover of by regular closed subsets of admits a countable subfamily that covers (resp., the closure of the union of whose members contains ). Obviously, every -set is an rc-Lindelöf set and every rc-Lindelöf set is an almost rc-Lindelöf set. It is also clear that a subset of a weak -space is rc-Lindelöf in if and only if it is almost rc-Lindelöf in .

Throughout this paper, (resp., , ) denotes the set of natural (resp., rational, real) numbers. For the concepts not defined here, we refer the reader to [18].

2. Slightly -continuous Mappings

This section is mainly devoted to study several properties of slightly -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) Let be a semi-continuous and almost open mapping. Then is weakly -irresolute.
(ii) Let be a semi-precontinuous and almost open mapping. Then is wrc-continuous.

Proof.. (i) Let be an open subset of . Since is almost open, then . Since is semi-continuous, then is semi-open, hence there exists an open subset of such that , therefore, . Thus is semi-open, and is weakly -irresolute.
(ii) Let be an open subset of . Since is almost open, then . Since is semi-precontinuous, then is semi-preopen, hence there exists a preopen subset of such that , therefore, . Thus is semi-preopen, and is wrc-continuous.

Corollary 2.2 (see [12]). Let be a semi-continuous and almost open mapping. Then is wrc-continuous.

Proposition 2.3. For a mapping , the following are equivalent: (i) is slightly continuous;(ii) whenever is semi-open in .

Proof.. Since every open set is semi-open, it suffices to show that (i)(ii). Let be a semi-open subset of . Then there exists an open subset of such that . Thus by (i), .

Proposition 2.4. Let be a slightly continuous mapping. Then the following are equivalent: (i) is weakly -irresolute;(ii) is rc-continuous.

Proof.. Since every regular closed set is semi-open, it suffices to show that (i)(ii). Let be a regular closed subset of . By (i), is semi-open, but is slightly continuous, so by Proposition 2.3, . Thus , that is, is closed, but is semi-open, so is regular closed. Hence is rc-continuous.

Corollary 2.5. Let be a slightly continuous, semi-continuous, and almost open mapping. Then is rc-continuous.

Proof.. Follows from Lemma 2.1(i) and Proposition 2.4.

Proposition 2.6. Let be a slightly continuous and semi-continuous mapping. Then for every open subset of .

Proof.. Let be an open subset of . Since is semi-continuous, it follows that is semi-open, but is slightly continuous, so it follows from Proposition 2.3 that . Thus .

The following corollary is a slight improvement of Corollary 2.5. This is because the closure of every semi-open set is regular closed.

Corollary 2.7. Let be a slightly continuous, semi-continuous, and almost open mapping. Then for every open subset of .

Proposition 2.8. Let be an rc-continuous mapping. If is rc-Lindelöf in , then is rc-Lindelöf in .

Proof.. Let be a cover of by regular closed subsets of . Then is a cover of by regular closed subsets of (as is rc-continuous). Since is rc-Lindelöf in , it follows that there exist such that , thus it follows that . Hence is rc-Lindelöf in .

Corollary 2.9 (see [19]). Let be a slightly continuous and weakly -irresolute mapping. If is rc-Lindelöf in , then is rc-Lindelöf in .

Proof.. Follows from Propositions 2.4 and 2.8.

Proposition 2.10 (see [20]). Let be a weakly continuous and almost open mapping. Then is rc-continuous.

Corollary 2.11. Let be a weakly continuous and almost open mapping. If is rc-Lindelöf in , then is rc-Lindelöf in .

Proof.. Follows from Propositions 2.10 and 2.8.

Now, we prove the following known result using a slight modification on the previous proof.

Proposition 2.12 (see [7]). Let be a slightly continuous and weakly -irresolute mapping. If is almost rc-Lindelöf in , then is almost rc-Lindelöf in .

Proof.. Let be a cover of by regular closed subsets of . Since is slightly continuous and weakly -irresolute, it follows from Proposition 2.4 that is rc-continuous and thus is a cover of by regular closed subsets of . Since is almost rc-Lindelöf in , it follows that there exist such that . Now, is regular closed and thus semi-open, but the arbitrary union of semi-open sets is semi-open, so is semi-open. Since is slightly continuous, it follows from Proposition 2.3 that . Hence is almost rc-Lindelöf in .

Definition 2.13. A mapping from a space into a space is said to be slightly -continuous if whenever is preopen in .

Proposition 2.14. For a mapping , the following are equivalent: (i) is slightly -continuous;(ii) whenever is semi-preopen in ;(iii) whenever is -open in .

Proof.. (i)(ii): Let be a semi-preopen subset of . Then there exists a preopen subset of such that . Thus by (i), .
(ii)(iii)(i): follow since every preopen set is -open and every -open set is semi-preopen.

Definition 2.15. A mapping is called brc-continuous if is -open for every regular closed subset of .

Clearly, every weakly -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. Let , , and . Then the identity mapping from onto is brc-continuous but not weakly -irresolute (observe that the regular closed subsets of are the members of , each of which is preopen and thus -open in . However, is not semi-open in .

Example 2.17. Let be the usual topology on the set of real numbers and , where . Then the identity mapping from onto is wrc-continuous but not brc-continuous (observe that the regular closed subsets of are the members of , each of which is semi-preopen in . However, is not -open in .

Proposition 2.18. Let be a slightly -continuous mapping. Then the following are equivalent:(i) is weakly -irresolute;(ii) is rc-continuous;(iii) is wrc-continuous;(iv) is brc-continuous.

Proof.. (ii)(i)(iv)(iii): follow since every regular closed set is semi-open, every semi-open set is -open and every -open set is semi-preopen.
(iii)(ii): let be a regular closed subset of . By (iii), is semi-preopen, but is slightly -continuous, so by Proposition 2.14, . Thus , that is, is closed, but is semi-preopen, so is regular closed. Hence is rc-continuous.

Corollary 2.19. Let be a slightly -continuous, semi-precontinuous, and almost open mapping. Then is rc-continuous.

Proof.. Follows from Lemma 2.1(ii) and Proposition 2.18.

Proposition 2.20. Let be a slightly -continuous and semi-precontinuous mapping. Then for every open subset of .

Proof.. Let be an open subset of . Since is semi-precontinuous, it follows that is semi-preopen, but is slightly -continuous, so it follows from Proposition 2.14 that . Thus .

Observing that the closure of every semi-preopen set is regular closed, the following corollary seems a slight improvement of Corollary 2.19.

Corollary 2.21. Let be a slightly -continuous, semi-precontinuous, and almost open mapping. Then for every open subset of .
Obviously, every continuous mapping is both semi-continuous and slightly -continuous and every slightly -continuous mapping is slightly continuous, the converses are, however, not true as the following two examples tell.

Example 2.22. Let , , and . Then the identity mapping from onto is slightly continuous and weakly -irresolute (observe that the regular closed subsets of are and ). However, it is not slightly -continuous (consider the preopen subset of ). We observe also that this is an example of a mapping that is both slightly continuous and semi-precontinuous but neither slightly -continuous nor semi-continuous (observe that , are both dense and thus preopen in . However, is not semi-open in ). This example also shows that the converses of Propositions 2.6 and 2.20 are not true.

Example 2.23. Let , , and . Then the identity mapping from onto is slightly -continuous (observe that the nonempty preopen subsets of are the supersets of ); it is, moreover, semi-continuous and almost open. However, it is not continuous.

Corollary 2.24. Let be a slightly -continuous and wrc-continuous mapping. If is rc-Lindelöf (resp., almost rc-Lindelöf) in , then is rc-Lindelöf (resp., almost rc-Lindelöf) in .

Proof.. We observe from Proposition 2.18 that a mapping that is both slightly -continuous and wrc-continuous is both slightly continuous and weakly -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. Let be a slightly -continuous, semi-precontinuous, and almost open mapping. If is rc-Lindelöf (resp., almost rc-Lindelöf) in , then is rc-Lindelöf (resp., almost rc-Lindelöf) in .

Proof.. Follows from Lemma 2.1(ii) and Corollary 2.24.

Remark 2.26. Since every dense set is preopen, one easily observes that if is a slightly -continuous mapping from a space onto a space , then maps dense subsets of onto dense subsets of .
Recall that a space is called submaximal (resp., strongly irresolvable) if every dense subset of is open (resp., semi-open), or equivalently if, every preopen subset of is open (resp., semi-open).

The following proposition is a direct consequence of Proposition 2.3.

Proposition 2.27. Let be a mapping from a strongly irresolvable space into a space . Then the following are equivalent: (i) is slightly -continuous;(ii) is slightly continuous.

3. Almost -open Mappings

Definition 3.1. A mapping from a space into a space is said to be semi-regular open (resp., semi- -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- -regular open. The converses are, however, not true as the following examples show.

Example 3.3. Let , , and . Then the identity mapping from onto is semi-regular open (observe that the regular open subsets of are the members of , each of which is semi-closed in ); it is, however, not regular open since is not regular open in .

Example 3.4. Let , , and . Then the identity mapping from onto is semi- -regular open (observe that and are preopen and thus semi-preopen in ); it is, however, not semi-regular open since is not semi-closed in .

Definition 3.5. A mapping from a space into a space is said to be almost -open if whenever is preopen in .

Proposition 3.6. For a mapping , the following are equivalent: (i) is almost open;(ii) whenever is semi-open in .

Proof.. Since every open set is semi-open, it suffices to show that (i)(ii). Let be a semi-open subset of . Then there exists an open subset of such that . Thus by (i), .

Proposition 3.7. For a mapping , the following are equivalent: (i) is almost -open;(ii) whenever is semi-preopen in ;(iii) whenever is -open in .

Proof.. (i)(ii): Let be a semi-preopen subset of . Then there exists a preopen subset of such that . Thus by (i), .
(ii)(iii)(i): follow since every preopen set is -open and every -open set is semi-preopen.

Remark 3.8. Since every open set is preopen, it is obvious that every almost -open mapping is almost open. However, the converse is not true as the following example tells.

Example 3.9. Let , , and . Then the identity mapping from onto is almost open and even regular open (observe that the regular open subsets of are and ); it is, however, not almost -open since is dense and thus preopen in but not dense in .

Proposition 3.10. For an almost -open mapping , the following are equivalent: (i) is semi- -regular open;(ii)semi-regular open;(iii)regular open.

Proof.. (i)(iii): Let be a regular open subset of . By assumption, is semi-preclosed, that is, is semi-preopen. By Proposition 3.7, . Thus and, therefore, , that is, , that is, is open, but is semi-preclosed, so is regular open.
(iii)(ii)(i): follow since every regular open mapping is semi-regular open and every semi-regular open mapping is semi- -regular open.

Proposition 3.11. For an almost open mapping , 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 be a regular open subset of . By assumption, is semi-closed, that is, is semi-open. By Proposition 3.6, . Thus and, therefore, , that is, , that is, is open, but is semi-closed, so is regular open.

Proposition 3.12 (see [19]). Let be an almost open and regular open mapping from a space onto a space . Then the following hold.
(i)If for each , is an -set in , then is almost rc-Lindelöf in whenever is almost rc-Lindelöf in .(ii)If for each , is rc-Lindelöf in , then is rc-Lindelöf in whenever is almost rc-Lindelöf in provided that is a weak -space.

Corollary 3.13. Let be an almost -open and semi- -regular open mapping from a space onto a space . Then the following hold.
(i)If for each , is an -set in , then is almost rc-Lindelöf in whenever is almost rc-Lindelöf in .(ii)If for each , is rc-Lindelöf in , then is rc-Lindelöf in whenever is almost rc-Lindelöf in provided that is a weak -space.

Proof.. We observe from Proposition 3.10 that a mapping that is both almost -open and semi- -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 if is an almost -open mapping from a space into a space , then the inverse image of a dense subset of is a dense subset of .

The following proposition is a direct consequence of Proposition 3.6.

Proposition 3.15. Let be a mapping from a space into a strongly irresolvable space . Then the following are equivalent: (i) is almost -open;(ii) is almost open.

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