Generalized Open Sets of Minkowski Space

We consider the Euclidean topology and s−topology on n−dimensional Minkowski space and investigate the interior and closure of the space cone and perforated space cone with respect to these topologies. In these regards, we determine that whether the space cone and perforated space cone are generalized open set or not. Moreover, we compare the generalized Euclidean topologies with the generalized s−topologies.


Introduction
In 1963, Zeeman explained that considering Euclidean topology, which is locally homogenous, on Minkowski space is not good enough since the null cone separates the spacelike vectors from the timelike ones in this space [25]. Zeeman suggested a new topology on Minkowski space M and introduced the finest topology on 4-dimensional Minkowski space that induces 1-dimensional Euclidean topology on any timelike line and 3-dimensional Euclidean topology on any spacelike hyperplane with their homeomorphisms groups generated by inhomogenous Lorentz group, translations and dilatations [25,26]. These topologies were called, respectively, time topology and space topology by [16]. Also, Nanda defined t−topology and s−topology which are, respectively, weaker than the Zeemans' time and space topology [16,17]. The order topology (Whiston called it Zeeman-order topology [24]) and A−topology on the Minkowski space were studied in [18,19], respectively. Dossena obtained the results that the Zeemans' topology as the finest topology on n−dimensional Minkowski space is separable, Hausdorff, non-normal, non-locally compact, non-Lindeloff and the first countable and moreover, he investigated 2−dimensional case [12]. The topological properties of t−topology and s−topology were studied and the characterizations of the compact sets of Minkowski space with these topologies were obtained by [2,3].
To the best of authors' knowledge generalized topology of the Minkowski space is not studied yet. In the last century the theory of generalized topology has been widely studied in the literature . The classes of subsets of a  topological space which are more and less nearly open sets have been represented throughout the [14,20,15,1,4,22,23,5]. Let (X, τ ) be a topological space and A be a subset of X. The closure and the interior of A are denoted by cl τ (A) and int τ (A), respectively. The subset A of (X, τ ) is called a regular-open set, if A = int τ (cl τ (A)) [22]. The finite union of regular open sets is said to be π−open in (X, τ ). The family of all regular open sets is denoted by RO (X), and RO (X) ⊂ τ . Also, a subset A of (X, τ ) is called α−open [20] (resp., β−open [1,4], semi-open [14], pre-open [15], and b−open [5] [8,13].
Some common properties of these well-known sets were considered and more general definitions were given by Császár in 1997, [6]. The remarkable class of studies on generalized open sets was given by Császár, [7,9,10,11]. The collection ζ of subsets of a non-empty X was called generalized topology by Császár such that, ∅ ∈ ζ, and arbitrary unions of elements of ζ belong to ζ. The collection of the generalized open sets defined in [6] constitutes a generalized topology. A set X with a generalized topology ζ was called as a generalized topological space, [7].
In these regards, we study generalized open subsets of Minkowski space endowed with the Euclidean topology and s−topology, respectively.

Preliminaries
The Lorentzian inner product of the vectors x = (x 0 , x 1 , ..., x n−1 ) and y = x i y i . The real vector space R n provided with the Lorentzian inner product g, which is symmetric, non-degenerate bilinear form, is called n−dimensional Minkowski space and denoted by M . Since g is an indefinite form, recall that x ∈ M can have three Lorentzian casual character as; it can be spacelike if g(x, x) > 0 or x = 0, timelike if g(x, x) < 0, and null (lightlike) if g(x, x) = 0 and x = 0, [21]. The group of linear operators T on M which leaves Lorentzian inner product g invariant as g( Soley Ersoy, Merve Bilgin, and İbrahim İnce 51 group. The sets are space cone, null (lightlike) cone, and time cone of x ∈ M , respectively. The Euclidean topology on n−dimensional Minkowski space M is the topology generated by the basis B = N E ε (x) : ε > 0, x ∈ M which will be called e−topology. M with the Euclidean topology will be denoted by M E and the elements of Euclidean topology on Minkowski space will be called e−open sets.
The s−topology on the n−dimensional Minkowski space M is defined by specifying the local base of neighborhoods at each point x ∈ M given by the collection Let us recall characters of cones in Minkowski space with respect to s−topology and e−topology from the following Lemmas given by [3]. Similar definitions and theorems are also available for t−topology on the n−dimensional Minkowski space M .

Some Generalized Open Sets of Minkowski Space
In this section, we introduce the interior and closure of some important subsets of Minkowski space with respect to the s−topology and e−topology. Then, we investigate that, whether these sets are generalized open set or not with respect to these topologies. Proof.
i. From the Lemma 2.1 and Lemma 2.
Conversely, let y ∈ (C S (x) ∪ C L (x)). Then y ∈ C S (x) or y ∈ C L (x). If y ∈ C S (x) then it is obvious that y ∈ cl e C S (x) . On the other case, if y ∈ C L (x) then there exists ∃z ∈ N E ε (y) for ∀ε > 0, such that z ∈ C S (x). Thus for ∀ε > 0, N E ε (y) ∩ C S (x) = ∅. This gives us y ∈ cl e C S (x) and (C S (x) ∪ C L (x)) ⊂ cl e C S (x) . This proves the assertion.
iii. Since C S (x) − {x} is an e−open set, this case is obvious. iv.
. But it is easy to see that y is not an interior point when y ∈ (C L (x) − {x}) or y = x since there isn't any This completes the proof. vi. This assertion is obvious since Proof. Suppose that G is arbitrary α − e−open set in M E . Then G ⊂ int e (cl e (int e (G))). From the Lemma 2.4, it follows that; ⇒ cl e (int e (G)) ⊃ cl s (int s (G)) ⇒ int e (cl e (int e (G))) ⊂ int s (cl s (int s (G))) ⇒ G ⊂ int e (cl e (int e (G))) ⊂ int s (cl s (int s (G))) ⇒ G ⊂ int s (cl s (int s (G)))