Continuity on r -near topological spaces

Abstract

In 2020, r-near topological spaces on Near Approximation Spaces were introduced by Atmaca [1]. In this study, we introduce the concept of continuity on r-near topological spaces and examine some properties of it.

Keywords

Near set continuity

In 2007, Peters [10, 11] introduced the near set theory as a generalization of the approach to classification of objects presented by Zdzislaw Pawlak [9]. Near set theory has practical applications in many areas such as feature selection [12], object recognition in images [4, 13], image processing [2], granular computing [14, 26] and in various forms of machine learning [8 , 18–20].

Recently, some researchers have introduced algebraic and topological structures of near sets. Applications of near sets in algebraic structures are given by İnan and Öztürk [5 –7]. They discussed the notion of Near groups on nearness approximation space and obtained some fundamental properties.

There are studies on the topological properties of nearness structures and nearness approximation spaces in the literature [24 , 27–29]. But topological structures of near sets were firstly introduced by Atmaca [1]. In that study, author defined r-near topology on a nearness approximation space and introduced some basic concepts of r-near topological spaces.

In this study, we introduce the concept of r-near continuous functions defined on a r-near topological spaces. Moreover we give some characterizations of the r-near continuity and support with examples and counter examples.

1 Introduction

In this section, we give properties and some definitions of near sets defined by Peters [10, 11].

Definition 1. [10] Let O be set of perceptual objects, F be a set of functions representing object features, x, x′ ∈ O and B ⊆ F. Then

_∼B = {(x, x′) ∈ O × O : φ (x) = φ (x′) , ∀φ ∈ B} is called the indiscernibility relation on O.

Definition 2. [10] Let O be set of perceptual objects, F be a set of functions representing object features, B ⊆ F and _∼Bbe indiscernibility relation. Then the triple FAS = (O, F, _∼B) is called Fundamental Approximation Space.

Definition 3. [10] Let (O, F, _∼B) be a Fundamental Approximation Space and A ⊆ O. Then B lower and upper approximations of A defined as

$B_{*} A = \underset{[x]_{B} \subseteq A}{\cup} [x]_{B}$

and

$B^{*} A = \underset{[x]_{B} \cap A \neq \emptyset}{\cup} [x]_{B}$ respectively, where [x] _B = {y ∈ O : (x, y) ∈ _∼B}.

Indiscernibility relation can be defined for each subset B_r, such that B_r ⊆ B ⊆ F and |B_r| = r. Let us denote this relation with _{∼B_r}. _{∼B_r}can form different decomposition of O for each r, where [x] _{B
_r} = {x′ : φ (x) = φ (x′) for all φ ∈ B_r}. Let N_r (B) = {O/_{∼B_r} : B_r ⊆ B} shows the set of all decompositons of O with respect to B.

Definition 4. [10] Let O be a set of perceptual objects, F be a set of functions representing object features and B ⊆ F. Then (O, F, _{∼B_r}, N_r (B)) is called Nearness Approximation Space.

Definition 5. [10] Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space and A ⊆ O. Then B_r lower and upper approximations of A defined as

$N_{r} (B)_{*} A = \underset{[x]_{B_{r}} \subseteq A}{\cup} [x]_{B_{r}}$

and

$N_{r} (B)^{*} A = \underset{[x]_{B_{r}} \cap A \neq \emptyset}{\cup} [x]_{B_{r}}$ .

A topological space can be thought of as a geometric space in which closeness is defined but cannot usually be measured by a numerical distance. Thus, an open set in a topological space is a set of elements close to each other. In [1], the author generalized the concept of closeness in topological spaces by using the concept of near set, and defined a new topological structure by adding elements that are weakly related to the elements of the set to open sets.

Definition 6. [1] Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. The family $τ_{r}^{*} = {N_{r} (B)^{*} G : G \in τ}$ is called r-near topology on X which generated by (O, F, _{∼B_r}, N_r (B)).

A member of $τ_{r}^{*}$ is called r-near open set in X and complement of a r-near open set is called r-near closed set.

The family of all r-near closed sets in X is denoted by $τ_{r}^{* k}$ . Throughout the paper, a r-near open set N_r (B) ^∗ G will be shown as $G_{r}^{*}$ for simplicity.

Definition 7. [1] Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O and (X, τ) be a topological space. If there exists a $G_{r}^{*} \in τ_{r}^{*}$ such that $x \in G \subseteq G_{r}^{*} \subseteq N$ , then N is called r-near neighborhood of x.

If N is r-near open set, then N is called r-near open neighborhood.

The family of all r-near neighborhood (r-near open neighborhood) of x is denoted by $N_{r}^{*} (x)$ ( $τ_{r}^{*} (x)$ ).

Definition 8. [1] Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A ⊆ X.

(1) The set ${cl}_{r}^{*} A : = \cap {F_{r}^{*} : F_{r}^{*}$ is a r-near closed set and $A \subseteq F_{r}^{*}}$ is called r-near closure of A .

(2) The set ${int}_{r}^{*} A : = \cup {G_{r}^{*} : G_{r}^{*}$ is a r-near open set and $G_{r}^{*} \subseteq A}$ is called r-near interior of A .

Proposition 1. [1] Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and r ≤ s ≤ |B|. Then followings are true;

(1) If x₀ ∈ X and $N \in N_{r}^{*} (x_{0})$ , then $N \in N_{s}^{*} (x_{0})$ .

(2) If A ⊆ B ⊆ X, then ${int}_{r}^{*} A \subseteq {int}_{r}^{*} B$

Definition 9. [1] Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and A ⊆ X. The set ${Core}_{r}^{*} A : = {x \in X : A \in N_{r}^{*} (x)$ } is called r-core of A.

Definition 10. [1] Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) be a topological space and (x_n) be a sequence in X. The sequence (x_n) is called r-near convergence to x₀ and, denoted by $x_{n} \overset{r}{\to} x_{0}$ , if for each $N \in N_{r}^{*} (x_{0})$ , there is $n_{0} \in ℕ$ such that n ≥ n₀ implies x_n ∈ N.

1.1 r-Near contiunity

Definition 11. Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) and (Y, σ) be topological spaces and f : X → Y be a function. Then f is called r-near continuous at x₀ ∈ X if there exist $U \in N_{r}^{*} (x_{0})$ such that f (U) ⊆ V for all V ∈ $N$ (f (x₀)).

If f is r-near continuous at x₀ for all x₀ ∈ X, then we say that f is r-near continuous on X.

Example 1. Let X = {a, b, c, d, e} , Y = {u, v, w, t, k}, B = {φ₁, φ₂, φ₃} be a set of functions representing object features, τ = {∅ , X, {a} , {c} , {a, b} , {a, c} , {c, d} , {a, b, c} , {a, c, d} , {a, b, c, d}} and σ = {∅ , Y, {u, v, w, t}}. Let assume that probe functions $φ_{i} : X \to ℝ$ defined as follows;

	a	b	c	d	e
φ ₁	1	1	0	1	0
φ ₂	1	2	1	1	0
φ ₃	1	1	2	0	0

Let us construct the equivalence classes for each r-combination where 1 ≤ r ≤ 3. Then equivalence classes are obtained as follows;

[a] _{φ₁} = {a, b, d}, [c] _{φ₁} = {c, e}

[a] _{φ₂} = {a, c, d}, [b] _{φ₂} = {b},[e] _{φ₂} = {e}

[a] _{φ₃} = {a, b}, [c] _{φ₃} = {c}, [d] _{φ₃} = {d, e},

[a] _{{φ₁,φ₂}} = {a, d}, [b] _{{φ₁,φ₂}} = {b}, [c] _{{φ₁,φ₂}} = {c}, [e] _{{φ₁,φ₂}} = {e}

[a] _{{φ₁,φ₃}} = {a, b}, [c] _{{φ₁,φ₃}} = {c}, [d] _{{φ₁,φ₃}} = {d}, [e] _{{φ₁,φ₃}} = {e}

[a] _{{φ₂,φ₃}} = {a}, [b] _{{φ₂,φ₃}} = {b}, [c] _{{φ₂,φ₃}} = {c}, [d] _{{φ₂,φ₃}} = {d}, [e] _{{φ₂,φ₃}} = {e}

[a] _{{φ₁,φ₂,φ₃}} = {a}, [b] _{{φ₁,φ₂,φ₃}} = {b}, [c] _{{φ₁,φ₂,φ₃}} = {c}, [d] _{{φ₁,φ₂,φ₃}} = {d}, [e] _{{φ₁,φ₂,φ₃}} = {e}

Thus we have

N₁ (B) = {{[a] _{φ₁}, [c] _{φ₁}}, {[a] _{φ₂}, [b] _{φ₂}, [e] _{φ₂}}, {[a] _{φ₃}, [c] _{φ₃}, [d] _{φ₃}}}

N₂ (B) = {{[a] _{{φ₁,φ₂}}, [b] _{{φ₁,φ₂}}, [c] _{{φ₁,φ₂}}, [e] _{{φ₁,φ₂}}}, {[a] _{{φ₁,φ₃}}, [c] _{{φ₁,φ₃}}, [d] _{{φ₁,φ₃}}, [e] _{{φ₁,φ₃}}}, {[a] _{{φ₂,φ₃}}, [b] _{{φ₂,φ₃}}, [c] _{{φ₂,φ₃}}, [d] _{{φ₂,φ₃}}, [e] _{{φ₂,φ₃}}}}

N₃ (B) = {{[a] _{{φ₁,φ₂,φ₃}}, [b] _{{φ₁,φ₂,φ₃}}, [c] _{{φ₁,φ₂,φ₃}}, [d] _{{φ₁,φ₂,φ₃}}, [e] _{{φ₁,φ₂,φ₃}}}}.

Hence $τ_{1}^{*} = {\emptyset, X, {a, b, c, d}, {a, c, d, e}}$ , $τ_{2}^{*} = {\emptyset, X, {c}, {a, b, d}, {a, c, d}, {a, b, c, d}}$ and $τ_{3}^{*} = τ$ .

Now let’s consider the function f : X → Y, f (a) = f (c) = u, f (b) = k, f (d) = f (e) = t. Since $N (f (a)) = N (u) = {Y, {u, v, w, t}}$ and $N_{1}^{*} (a) = {{a, b, c, d}, {a, c, d, e}, X}$ , f is 1-near continuous at a. Similarly, f is 1-near continuous at b,c,d and e. But, f is neither 2-near continuous nor 3-near continuous.

Theorem 1. Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) and (Y, σ) be topological spaces. Then, the following are equivalent for the function f : X → Y and x₀ ∈ X.

(1) f is r-near continuous at x₀,

(2) For each $V \in N (f (x_{0})$ , then there exist $U \in N_{r}^{*} (x_{0})$ such that U ⊆ f^-1 (V),

(3) For each $V \in N (f (x_{0})$ , $f^{- 1} (V) \in N_{r}^{*} (x_{0})$ .

Proof. (1) ⇒ (2) Obvious from the definiton.

(2) ⇒ (3) Let $V \in N (f (x_{0}))$ . From (2) there exist $U \in N_{r}^{*} (x_{0})$ such that U ⊆ f^-1 (V). Since $U \in N_{r}^{*} (x_{0})$ , $x_{0} \in G \subseteq G_{r}^{*} \subseteq U \subseteq f^{- 1} (V)$ for a G ∈ τ. Hence $f^{- 1} (V) \in N_{r}^{*} (x_{0})$ .

(3) ⇒ (1) Let $V \in N (f (x_{0}))$ . From (3) $f^{- 1} (V) \in N_{r}^{*} (x_{0})$ . If we choose U = f ^-1 (V), f (U) = f (f^-1 (V)) ⊆ V. This shows that f is r-near continuous at x₀. ■

Theorem 2. Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) and (Y, σ) be topological spaces and f : X → Y be a function. f is r-near continuous at x₀ if and only if $x_{0} \in {Core}_{r}^{*} (f^{- 1} (G))$ for all neighborhood G of f (x₀).

Proof. (⇒ :) Let f be r-near continuous at x₀ and G be a neighborhood of f (x₀). Since f is r-near continuous at x₀, then there exists a r-near neighborhood U of x₀ such that f (U) ⊆ G and so we obtain an open set H which satisfies $x_{0} \in H \subseteq H_{r}^{*} \subseteq U \subseteq f^{- 1} (G)$ . Consequently, we have that $x_{0} \in {Core}_{r}^{*} (f^{- 1} (G))$ .

(⟸ :) Let G be a neighborhood of f (x₀) and $x_{0} \in {Core}_{r}^{*} (f^{- 1} (G))$ . Then there exists U ∈ τ such that $x_{0} \in U \subseteq U_{r}^{*} \subseteq f^{- 1} (G)$ . Since $U_{r}^{*} \in N_{r}^{*} (x_{0})$ and $f (U_{r}^{*}) \subseteq G$ , we have that f is r-near continuous at x₀. ■

Theorem 3. Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) and (Y, σ) be topological spaces. Then, the following are true for the r-near continuous function f : X → Y.

(1) For each G ∈ σ, $f^{- 1} (G) \in τ_{r}^{*}$ .

(2) For each closed set F in Y, f^-1 (F) r-near closed set in X.

Proof. (1) Let G ∈ σ and x ∈ f^-1 (G) Since f (x) ∈ G and $G \in N (f (x))$ , by r-near continuity of f, there exist $H \in N_{r}^{*} (x)$ such that f (H) ⊆ G . On the other hand, there exists U ∈ τ such that $x \in U \subseteq U_{r}^{*} \subseteq H$ . This shows that $x \in U_{r}^{*} \subseteq f^{- 1} (G)$ . Consequently, we have that $f^{- 1} (G) \in τ_{r}^{*}$ .

(2) It is a result of equality X - f^-1 (F) = f^-1 (Y - F) and (1). ■

The converse of the above theorem is not true in general.

Example 2. Let us consider the Example 1 and σ = {∅ , Y,{u, v}}. If we take the function f : X → Y defined by f (a) = f (b) = u, f (c) = w, f (d) = v and f (e) = k, then f^-1 (G)∈ $τ_{2}^{*}$ for all G ∈ σ. But f is not is not 2-near continuous at d. Because there is no any 2-near neighborhood U of d such that f (U) ⊆ {u, v} for ${u, v} \in N (f (d))$ .

Remark 1. Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) and (Y, σ) be topological spaces and f : X → Y be a function. If $f^{- 1} (G) \notin τ_{r}^{*}$ for all G ∈ σ, then f is not r-near continuous.

Theorem 4. Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) and (Y, σ) be topological spaces. If f : X → Y is r-near continuous at x₀ and s < r, then f is s-near continuous at x₀.

Proof. Let f be r-near continuous at x₀ and $V \in N (f (x_{0}))$ . Then there exists $U \in N_{r}^{*} (x_{0})$ such that f (U) ⊆ V. By Propositon 1 (1), $U \in N_{s}^{*} (x_{0})$ and hence f is s-near continuous at x₀. ■

Theorem 5. Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) and (Y, σ) be topological spaces and f : X → Y be a function.If f is r-near continuous,then $f^{- 1} (intB) \subseteq {int}_{r}^{*} (f^{- 1} (B))$ .

Proof. Let f be a r-near continuous function and B ⊆ Y. Since intB ∈ σ, by Theorem 3, $f^{- 1} (intB) \in τ_{r}^{*}$ and so we have $f^{- 1} (intB) = {int}_{r}^{*} (f^{- 1} (intB)) \subseteq {int}_{r}^{*} (f^{- 1} (B))$ . ■

Definition 12. Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) and (Y, σ) be topological spaces and f : X → Y be a function. f is called r-near sequentally continuous at x₀ if f (x_n) → f (x₀) whenever $x_{n} \overset{r}{\to} x_{0}$ .

Theorem 6. Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, X ⊆ O, (X, τ) and (Y, σ) be topological spaces and f : X → Y be a function. If f is r-near continuous at x₀, then f is r-near sequentally continuous at x₀.

Proof. Let f be a r-near continuous at x₀ and (x_n) be a sequence in X with $x_{n} \overset{r}{\to} x_{0}$ . Now, let us consider V as neighborhood of f (x₀). Since f is r-near continuous, there exists $U \in N_{r}^{*} (x_{0})$ such that f (U) ⊆ V. Since $x_{n} \overset{r}{\to} x_{0}$ , then there exists $n_{0} \in ℕ$ such that x_n ∈ U for all n ≥ n₀. Therefore, we have f (x_n) ∈ f (U) ⊆ V for all n ≥ n₀ and hence f (x_n) → f (x₀). ■

The converse of the above theorem is not true in general.

Example 3. Let (O, F, _{∼B_r}, N_r (B)) be a Nearness Approximation Space, where $O = ℝ$ and $F = {φ ∣ φ : ℝ \to ℝ$ is injective}. Let us consider cocountable and usual topologies on $ℝ$ denoted by τ_cc and τ_e, respectively. It is known that any sequence (x_n) converging to point a in the topological space $(ℝ, τ_{cc})$ is in the form (x_n) = (x₁, x₂, . . . , x_{n
₀}, a, a, a, . . .). Then the function $I : (ℝ, τ_{cc}) \to (ℝ, τ_{e})$ defined by I (x) = x is 1-near sequentally continuous at a. Because $(τ_{cc})_{1}^{*}$ is equal to τ_cc. But I is not 1-near continuous at a.

Conclusion. We introduced notion of continuity, one of the most important concepts of topological spaces, in the context of r-near topological spaces. We hope that the concepts given in this study will be a guide for new theoretical and practical studies on Nearness Approximation Spaces.

The first author in this work is supported by the Scientific Research Project Fund of Sivas Cumhuriyet University under project number F-600.

Footnotes

Acknowledgment

The authors gratefully thank to the Referee for the constructive comments and recommendations which definitely help to improve the readability and quality of the paper.

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