Weak Forms of Continuity and Associated Properties

We introduce slightly 𝑝-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.


Introduction and preliminaries
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 wrccontinuous.In 7 , it is shown that the properties semi-continuous and slightly continuous are independent of each other.
A space X is called a weak P -space 13 if for each countable family 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 space X is called rc-Lindel öf 14 if every regular closed cover of X has a countable subcover, and called almost rc-Lindel öf 15 if every regular closed cover of X has a countable subfamily whose union is dense in X.
A subset A 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-Lindel öf set in X resp., an almost rc-Lindel öf 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-Lindel öf set and every rc-Lindel öf set is an almost rc-Lindel öf set.It is also clear that a subset A of a weak P -space X is rc-Lindel öf in X if and only if it is almost rc-Lindel öf 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 .

Slightly p-continuous mappings
This section is mainly devoted to study several properties of slightly p-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.ii Let f : X → Y be a semi-precontinuous and almost open mapping.Then f is wrccontinuous.
Proposition 2.4.Let f : X → Y be a slightly continuous mapping.Then the following are equivalent: Proposition 2.6.Let f : X → Y be a slightly continuous and semi-continuous mapping.Then 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 f : X → Y be a slightly continuous, semi-continuous, and almost open mapping. Then
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 f : X → Y be a slightly continuous and weakly θ-irresolute mapping.
Proof.Let {U α : 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 U α : α ∈ Λ} is a cover of A by regular closed subsets of X.Since A is almost rc-Lindel öf in X, it follows that there exist Definition 2.13.A mapping f 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 mapping f : X → Y , the following are equivalent: Proof.i → ii : Let U be a semi-preopen subset of X.Then there exists a preopen subset V of Clearly, every weakly θ-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 2.17.Let τ u 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, τ u 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, τ u .However, A is not b-open in R, τ u .Proposition 2.18.Let f : X → Y be a slightly p-continuous mapping.Then the following are equivalent: 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, Proposition 2.20.Let f : X → Y be a slightly p-continuous and semi-precontinuous mapping.Then Proof.Let U 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 Observing that the closure of every semi-preopen set is regular closed, the following corollary seems a slight improvement of Corollary 2.19.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 International Journal of Mathematics and Mathematical Sciences 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.

Corollary 2.21. Let f : X → Y be a slightly p-continuous, semi-precontinuous, and almost open mapping. Then
Example 2.23.Let X {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. Let f : X → Y be a slightly p-continuous and wrc-continuous mapping. If
Proof.We observe from Proposition 2.18 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. Let f : X → Y be a slightly p-continuous, semi-precontinuous, and almost open mapping. If
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 f is a slightly pcontinuous mapping from a space X onto a space Y , then f maps dense subsets of X onto dense subsets of Y .
Recall that a space X 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 Proposition 2.3.
Proposition 2.27.Let f : X → Y be a mapping from a strongly irresolvable space X into a space Y .Then the following are equivalent: ii f is slightly continuous.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 * .

Almost p-open mappings
Proposition 3.6.For a mapping f : X → Y , the following are equivalent: Proposition 3.7.For a mapping f : X → Y , the following are equivalent: Proof.i → ii : Let U be a semi-preopen subset of Y .Then there exists a preopen subset iii regular open.ii If for each y ∈ 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.i If for each y ∈ 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 .

International Journal of Mathematics and Mathematical Sciences
ii If for each y ∈ 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 Proposition 3.10 that a mapping that is both almost p-open and semip-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 f is an almost popen 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 Proposition 3.6.
Proposition 3.15.Let f : X → Y be a mapping from a space X into a strongly irresolvable space Y .Then the following are equivalent: ii f is almost open.

A
subset A of a space X is called regular open if A Int A, and regular closed if X \ A is regular open, or equivalently, if A Int A. It is well known that a subset A of a space X is regular open if and only if A Int F, 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 ⊂ Int A resp., A ⊂ Int A, A ⊂ Int A, A ⊂ Int A ∪ Int A .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 mapping f 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

Lemma 2 . 1 .
i Let f : X → Y be a semi-continuous and almost open mapping.Then f is weakly θ-irresolute.

Corollary 2 . 5 .
Let f : X → Y be a slightly continuous, semi-continuous, and almost open mapping.Then f is rc-continuous.Proof.Follows from Lemma 2.1 i and Proposition 2.4.
follow since every regular closed set is semi-open, every semiopen set is b-open and every b-open set is semi-preopen.

Corollary 2 . 19 .
Let f : X → Y be a slightly p-continuous, semi-precontinuous, and almost open mapping.Then f is rc-continuous.Proof.Follows from Lemma 2.1 ii and Proposition 2.18.
Proof.Since every open set is semi-open, it suffices to show that i → ii .Let U be a semiopen subset of Y .Then there exists an open subset

Remark 3 . 8 .Proposition 3 . 10 .
since every preopen set is b-open and every b-open set is semipreopen.Since every open set is preopen, it is obvious that every almost p-open mapping is almost open.However, the converse is not true as the following example tells.Example 3.9.Let X {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 .For an almost p-open mapping f : X → Y , the following are equivalent: i f is semi-p-regular open; ii semi-regular open;

Proposition 3 . 11 .
follow since every regular open mapping is semi-regular open and every semi-regular open mapping is semi-p-regular open.For an almost open mapping f : 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 so f A is regular open.Proposition 3.12 see 19 .Let f be an almost open and regular open mapping from a space X onto a space Y .Then the following hold.iIf for each y ∈ 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 .

Corollary 3 . 13 .
Let f be an almost p-open and semi-p-regular open mapping from a space X onto a space Y .Then the following hold.

Corollary 2.2 see
and f is wrc-continuous.12 .Let f : X → Y be a semi-continuous and almost open mapping.Then f is wrc-continuous.
Proposition 2.3.For a mapping f : X → Y , the following are equivalent: for every open subset U of Y .Obviously, every continuous mapping is both semi-continuous and slightly p-continuous and every slightly p-continuous mapping is slightly continuous, the converses are, however, not true as the following two examples tell.
Definition 3.1.A mapping f 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., semipreclosed subsets.Since every regular open set is semi-closed and every semi-closed set is semipreclosed, 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.Let X {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 * .