Further Remarks On Somewhere Dense Sets

In this article, we prove that a topological space X is strongly hyperconnected iff any somewhere dense set in X is open, in addition we investigate some conditions that make sets somewhere dense in subspaces, finally, we show that any topological space defined on infinite set X has SD-cover with no proper subcover.

. The concept of somewhere dense set was due to Pugh [6], where a set A is somewhere dense if the interior of its closure is non-empty, clearly somewhere dense set is a generalization of both open set and dense set. In 2017, Alshami [7] provided the properties of somewhere dense sets, and he introduced the axiom of S 1 , then with Noiri they defined the notion of SD-cover and use it to introduced compactness and lindelöfness via somewhere dense sets, see [8,9]. A space is hyperconnected [10], if every two non-empty open sets intersect; equivalently if any non-empty open set is dense, while a space is submaximal [11] if any dense set is open, and in 1994 Rose and Mahmoud [12] showed that a space is submaximal if and only if every preopen set is open, for more details see [13,14,15,16,17,18,19,20,21]. Recently, Alshami [7,8] defined strongly hyperconnected space as a hyperconnected submaximal space, and he charactrized this space using the notion of somewhere dense sets [8]. The main goal of this article, is to continue studying further properties on somewhere dense sets, and imporve some of the results given by Alshami and Noiri [7,8] regarding strongly hyperconnected space. Here we give solutions to the following questions, equiped with some examples: Question 1. Find the necessary and sufficient condition under which every somewhere dense set is open?. Question 2. If (X,τ) is a topological space and Y⊆ X: find conditions under which set in the subspace Y is somewhere dense in X?. Question 3. If (X,τ) is a topological space where X is infinite set: find a cover for X by somewhere dense sets (SD-cover) which has no proper subcover. The article is divided into four sections: somewhere dense sets in strongly hyperconnected space, somewhere dense sets in subspaces, SD-covers and conclusion. Throughout this article X or (X,τ) JOPAS Vol.21 No. 1 20 2 2 47 represents topological space and for a subset A of a space X, A and A ο denote the closure and the interior of A; respectively. Moreoever, X/A (or A c ), A/B and P(X) denote the complement of the set A in X, the difference of A and B, and the power set of X; respectively, while ~ denotes the equivalence relation, and χ 0 , χ 1 are the cardinality of the natural numbers ℕ and the real numbers ℝ; respectively.

Somewhere Dense Sets In Strongly Hyperconnected Spaces
This section, consists basic definitions, theorems and some properties regarding somewhere dense sets needed in this work, and then we give a complete answer for question 1, by studying the statament when any somewhere dense set is open. Definition 2.1. [7] A subset B of a topological space (X,τ) is called somewhere dense (briefly s-dense) if the interior of its closure is nonempty, i.e. B ο ≠ . The complement of s-dense set is called closed somewhere dense (briefly cs-dense), and the collection of all s-dense sets in X is denoted by S(τ). Corollary 2.1. [7] In a topological space (X,τ), we have : i. any open set is s-dense.
ii. any dense set is s-dense iii. any set in X that contains a s-dense set is s-dense. Theorem 2.1. [7] Every subset of a space (X,τ) is s-dense or csdense. Theorem 2.2. For a non-discrete topological space (X,τ), the following are equivalent: 1) X is strongly hyperconnected space.
2) X is submaximal and hyperconnected space. i. If (X,τ) is a trivial topological space where X has more than one element, then X is not strongly hyperconnected since S(τ)=P(X)/{ }=D(τ).

Somewhere Dense Sets In Subspaces
Here we answer question 2 by investigating some conditions in topological space X that make a set in a subspace somewhere dense in X. Corollary 3.1. [11] Let (X,τ) be a topological space, Y be a subspace of X and A⊆Y, then: respect to the subspace Y). Lemma 3.1. Let (X,τ) be a topological space, and Y be a subspace of X, then: (1) and (2). Theorem 3.1. Let (X,τ) be a topological space, Y be a subspace of X, A⊆Y then: 1) If Y is closed and A is s-dense in X, then A is s-dense in Y.

2) If Y is open and A is s-dense in Y, then A is s-dense in X. 3) If Y is clopen, then A is s-dense in Y iff
A is s-dense in X.

PROOF.
1) Since A is s-dense in X, we have A ο ≠ . Y is closed then A⊆Y, so A ο ⊆ (A ) ο Y and from the prevoius lemma (1) we thus A ο ≠ , i.e. A is s-dense in X. (1) and (2). Example 3.1. In the space (ℝ,τ) where τ ={U⊆ℝ: 0∈U}∪{ }, we have S(ℝ) =τ/{ }, so if Y= ℝ/{0} and A={1}, then Y is closed and A is s-dense in Y, while A is not s-dense in X since the relative topology on Y is the discrete topology. Corollary 3.2. Let (X,τ) be a topological space, and A be a subset of X with non-empty interior, then A is s-dense. PROOF. Since A⊆ A we have A ο ⊆ A ο , and A ο =A implies that A ο ≠ . Corollary 3.3. Let (X,τ) be a topological space, and A be a subset of X with non-empty interior, then A is s-dense in any subspace Y containing A.

PROOF. Suppose Y is a subspace of X, and A⊆Y then
≠ , thus A is s-dense in Y. Definition 3.1. [22] A subset B of a space (X, ) is called regular closed (briefly r-closed) if B=B ο . Note that, any r-closed set is closed. Theorem 3.2. Let (X,τ) be a topological space, Y be an r-closed subspace of X, A⊆Y, then A is s-dense in Y iff A is s-dense in X. 48 ⟸ Direct from theorem (3.1), since any r-closed set is closed.
⟹ Suppose A is s-dense in Y, then ((A Y ) ο Y ) ≠ , and since Y is r- then (A) ο Y ∩ Y ο = , and since (A) ο Y ≠ ϕ, there exists an open set W such that W∩ Y c ≠ , W∩ A ≠ and W∩ Y ⊆ A , therefore W∩ Y ⊆ (A) ο Y . Now Y is r-closed set, so we have Y ο ≠ ϕ, and since which is a contradiction since W∩ A ≠ and A ⊆Y. Hence A ο ≠ ϕ, thus A is s-dense in X.

SD-Covers
In the present section, we answer question 3 by proving that any topological space defined on infinite set X has a cover by somewhere dense sets with no proper subcover. Definition 4.1. [9] If (X,τ) is a topological space, then a cover for X by s-dense subsets is called SD-cover for X. Remark 4.1. Any cover for a space X is SD-cover, but the converse is not true. Examples 4.1. i.
If (X,τ) is a trivial topological space, then S(τ)=P(X)/{ }, so when X is uncountable then the collection of all singeltons is SD-cover for X with no countable subcover. ii.
If (X,τ) is the cofinite topological space, then S(τ)={U⊆X: U is infinite}, so if X = A⋃B where A is countable, B is uncountable and A∩B = ϕ, then {A⋃{x} } x∈B is an SDcover for X with no countable subcover. Thus |Y|=|B|=|B c |=χ n+1 . This complete the prove. Theorem 4.1. If (X,τ) is a topological space where X is infinite, then (X,τ) has a SD-cover with no proper subcover. PROOF. From lemma (4.1) there exists a subset A of X such that |X|=|A|=|A c |. So A and A c are infinite, and form theorem (2.1) at least A or A c is s-dense. Suppose A is s-dense, then from corollary (2.1.(iii)) A⋃{x} is also s-dense for any x ∈ A c , hence {A⋃{x} } x∈A c is SD-cover for X with no proper subcover. Similarly in the case when A c is s-dense, we have {A c ⋃{x} } x∈A is SD-cover for X with no proper subcover.

Conclusion
In this article, we investigate some further topological properties on somewhere dense sets, and we have obtained few results; as follows: If (X,τ) is a topological space; then any somewhere dense set in X is open if and only if X is strongly hyperconnected, and if X is infinite set, then there exists a cover for X by somewhere dense sets with no proper subcover. Moreover, if Y is a subspace and A⊆Y⊆X; then: In the case when the subspace Y is open (closed) in X, if A is somewhere dense in Y (X), then A is somewhere dense in X (Y), while in the case when the subspace Y is regular closed in X, A is somewhere dense in Y if and only if A is somewhere dense in X.