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ISRN Mathematical Physics
VolumeΒ 2012Β (2012), Article IDΒ 896156, 11 pages
http://dx.doi.org/10.5402/2012/896156
Research Article

A Study of Non-Euclidean s-Topology

Department of Mathematics, Faculty of Science, Dayalbagh Educational Institute (Deemed University), Dayalbagh, Agra 282 110, India

Received 30 April 2012; Accepted 28 May 2012

Academic Editors: G.Β Cleaver and M.Β Znojil

Copyright Β© 2012 Gunjan Agrawal and Sampada Shrivastava. 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

The present paper focuses on the characterization of compact sets of Minkowski space with a non-Euclidean 𝑠-topology which is defined in terms of Lorentz metric. As an application of this study, it is proved that the 2-dimensional Minkowski space with 𝑠-topology is not simply connected. Also, it is obtained that the 𝑛-dimensional Minkowski space with 𝑠-topology is separable, first countable, path-connected, nonregular, nonmetrizable, nonsecond countable, noncompact, and non-LindelΓΆf.

1. Introduction

Non-Euclidean topologies on 4-dimensional Minkowski space were first introduced by Zeeman [1] in 1967. These topologies include fine, space topology [2], time topology [3], 𝑑-topology [3], and 𝑠-topology [3]. Studying the homeomorphism group of 4-dimensional Minkowski space with fine topology, Zeeman in his paper [1] mentioned that it is Hausdorff, connected, locally connected space that is not normal, not locally compact and not first countable. His results were interesting both topologically and physically, because its homeomorphism group was the group generated by the Lorentz group, translations and dilatations which was exactly the one physicists would want it to be. Continuing the study of non-Euclidean topologies, Nanda in his papers [2, 3] mentioned that the 4-dimensional Minkowski space, with the space topology, is Hausdorff but neither normal nor locally compact nor second countable and that with each of the 𝑑-topology and 𝑠-topology is a nonnormal, noncompact Hausdorff space besides proving that the homeomorphism group of 4-dimensional Minkowski space with space, 𝑑 and 𝑠-topologies, is generated by the Lorentz group, translations, and dilatations. Further, Nanda and Panda [4] introduced the notion of a non-Euclidean topology, namely, order topology, and obtained that it is a non-compact, non-Hausdorff, locally connected, connected, path connected, simply connected space. In 2007, Dossena [5] proved that the 𝑛-dimensional Minkowski space, 𝑛>1, with the fine topology is separable, Hausdorff, nonnormal, nonlocally compact, non-LindelΓΆff and nonfirst countable. He further obtained that 2-dimensional Minkowski space with fine topology is path connected but not simply connected and characterized its compact sets. Quite recently, in 2009, Agrawal and Shrivastava [6] obtained a characterization for compact sets of Minkowski space with 𝑑-topology besides studying its topological properties. It may be noted that 𝑑-topology on 4-dimensional Minkowski space is same as that of the well-known path topology on strongly causal spacetime proposed by Hawking et al. in 1976 [7].

The present paper explores the 𝑠-topology on 𝑛-dimensional Minkowski space. Section-wise description of the work carried out in this paper is given below.

Beginning with an introduction, necessary notation and preliminaries have been provided in Section 2. In Section 3, it is proved that the 𝑠-topology on 𝑛-dimensional Minkowski space is strictly finer than the Euclidean topology by studying open sets, closed sets and subspace topologies on certain subsets of Minkowski space with 𝑠-topology. Topological properties of Minkowski space with 𝑠-topology are dealt in Sections 4, 5, and 6. In Section 7, compact subsets of Minkowski space with 𝑠-topology have been characterized. As a consequence of this study, it is proved that 2-dimensional Minkowski space with 𝑠-topology is not simply connected. Finally, Section 8 concludes the paper.

2. Notation and Preliminaries

Let Ξ› denote an indexing set while 𝑅, 𝑁, and 𝐾 denote the set of real, natural and rational numbers, respectively. To avoid any confusion later, we mention here that the symbol 𝑄, in this paper, denotes the indefinite characteristic quadratic form. For a subset 𝐴 of a set 𝑋, π‘‹βˆ’π΄ denotes the complement of 𝐴 in 𝑋. For π‘₯,π‘¦βˆˆπ‘…π‘›, let 𝑑𝐸(π‘₯,𝑦) be the Euclidean distance between π‘₯ and 𝑦. For πœ–>0,π‘πΈπœ–(π‘₯) denotes the πœ–-Euclidean neighborhood about π‘₯ given by the set {π‘¦βˆˆπ‘…π‘›βˆΆπ‘‘πΈ(π‘₯,𝑦)<πœ–}. For π‘₯,π‘¦βˆˆπ‘…π‘›, let [π‘₯,𝑦] denote the line segment joining π‘₯ and 𝑦.

The 𝑛-dimensional Minkowski Space, denoted by 𝑀, is the 𝑛-dimensional real vector space 𝑅𝑛 with a bilinear form π‘”βˆΆπ‘…π‘›Γ—π‘…π‘›β†’π‘…, satisfying the following properties:(i)for all π‘₯,π‘¦βˆˆπ‘…π‘›, 𝑔(π‘₯,𝑦)=𝑔(𝑦,π‘₯), that is, the bilinear form is symmetric(ii)if for all π‘¦βˆˆπ‘…π‘›, 𝑔(π‘₯,𝑦)=0, then π‘₯=0, that is, the bilinear form is nondegenerate, and(iii)there exists a basis {𝑒0,𝑒1,…,π‘’π‘›βˆ’1} for 𝑅𝑛 with 𝑔𝑒𝑖,𝑒𝑗=⎧βŽͺ⎨βŽͺ⎩1if𝑖=𝑗=0βˆ’1if𝑖=𝑗=1,2,…,π‘›βˆ’10if𝑖≠𝑗.(2.1)The bilinear form 𝑔 is called the Lorentz inner product.

Elements of 𝑀 are referred to as events. If βˆ‘π‘₯β‰‘π‘›βˆ’1𝑖=0π‘₯𝑖𝑒𝑖 is an event, then the coordinate π‘₯0 is called the time component and the coordinates π‘₯1,…,π‘₯π‘›βˆ’1 are called the spatial components of π‘₯ relative to the basis {𝑒0,𝑒1,…,π‘’π‘›βˆ’1}. In terms of components, the Lorentz inner product 𝑔(π‘₯,𝑦) of two events βˆ‘π‘₯β‰‘π‘›βˆ’1𝑖=0π‘₯𝑖𝑒𝑖 and βˆ‘π‘¦β‰‘π‘›βˆ’1𝑖=0𝑦𝑖𝑒𝑖 is defined by π‘₯0𝑦0βˆ’βˆ‘π‘›βˆ’1𝑖=1π‘₯𝑖𝑦𝑖. Lorentz inner product induces an indefinite characteristic quadratic form 𝑄 on 𝑀 given by 𝑄(π‘₯)=𝑔(π‘₯,π‘₯). Thus 𝑄(π‘₯)=π‘₯20βˆ’βˆ‘π‘›βˆ’1𝑖=1π‘₯2𝑖. The group of all linear operators 𝑇 on 𝑀 which leave the quadratic form 𝑄 invariant, that is, 𝑄(π‘₯)=𝑄(𝑇(π‘₯)), for all π‘₯βˆˆπ‘€, is called the Lorentz group.

A event π‘₯βˆˆπ‘€ is called spacelike, lightlike (also called null) or timelike vector according as 𝑄(π‘₯) is negative, zero, or positive. The sets 𝐢𝑆(π‘₯)={π‘¦βˆˆπ‘€βˆΆπ‘¦=π‘₯ or 𝑄(π‘¦βˆ’π‘₯)<0},𝐢𝐿(π‘₯) = {π‘¦βˆˆπ‘€βˆΆπ‘„(π‘¦βˆ’π‘₯)=0}, 𝐢𝑇(π‘₯)={π‘¦βˆˆπ‘€βˆΆπ‘¦=π‘₯ or 𝑄(π‘¦βˆ’π‘₯)>0} are likewise, respectively called the space cone, light cone (or null cone), and time cone at π‘₯. For given π‘₯,π‘¦βˆˆπ‘€, the set {π‘₯+𝑑(π‘¦βˆ’π‘₯)|π‘‘βˆˆπ‘…} is called a spacelike straight line or light ray or timelike straight line joining π‘₯ and 𝑦 according as 𝑄(π‘¦βˆ’π‘₯) is negative or zero or positive. For further details, we refer to [8].

The Euclidean topology on the 𝑛-dimensional Minkowski space 𝑀 is the topology generated by the basis 𝐡={π‘πΈπœ–(π‘₯)βˆΆπœ–>0,π‘₯βˆˆπ‘€}. 𝑀 with the Euclidean topology will be denoted by 𝑀𝐸.

The 𝑠-topology on the 𝑛-dimensional Minkowski space 𝑀 is defined by specifying the local base of neighborhoods at each point of π‘₯βˆˆπ‘€ given by the collection 𝒩(π‘₯)={π‘π‘ πœ–(π‘₯)βˆΆπœ–>0}, where π‘π‘ πœ–(π‘₯)=π‘πΈπœ–(π‘₯)βˆ©πΆπ‘†(π‘₯). We call π‘π‘ πœ–(π‘₯) the 𝑠-neighborhood of radius πœ–. 𝑀 endowed with 𝑠-topology is denoted by 𝑀𝑠. For a subset 𝐴 of 𝑀, 𝐴𝑠 (𝐴𝐸) denotes the subspace 𝐴 of 𝑀𝑠 (𝑀𝐸).

3. Important Subsets and Subspaces of 𝑀𝑠

In this section, besides proving that the 𝑠-topology on 𝑀 is strictly finer than the Euclidean topology on 𝑀, important subsets and subspaces of 𝑀𝑠, which will use in the following sections, are studied.

Lemma 3.1. Let 𝑀 be the 𝑛-dimensional Minkowski space and π‘₯βˆˆπ‘€. Then 𝐢𝑇(π‘₯)βˆ’{π‘₯}, and 𝐢𝑆(π‘₯)βˆ’{π‘₯} are open in 𝑀𝐸 and 𝐢𝐿(π‘₯) is closed in 𝑀𝐸.

Proof. For π‘’βˆˆπ‘€πΈ, define π‘“βˆΆπ‘€πΈβ†’π‘… by 𝑓(𝑒)=(𝑒0βˆ’π‘₯0)2βˆ’βˆ‘π‘›βˆ’1𝑖=1(π‘’π‘–βˆ’π‘₯𝑖)2. Then 𝑓 is continuous and π‘“βˆ’1(0,∞)=𝐢𝑇(π‘₯)βˆ’{π‘₯},π‘“βˆ’1(βˆ’βˆž,0)=𝐢𝑆(π‘₯)βˆ’{π‘₯} and π‘“βˆ’1{0}=𝐢𝐿(π‘₯). Since (0,∞) and (βˆ’βˆž,0) are open and {0} is closed in 𝑀𝐸, the results follow.

In the following lemma, it is proved that the 𝑠-neighborhoods are open in 𝑀𝑠.

Lemma 3.2. Let 𝑀 be the 𝑛-dimensional Minkowski space and π‘₯βˆˆπ‘€. Then π‘π‘ πœ–(π‘₯),πœ–>0 is open in 𝑀𝑠.

Proof. It is sufficient to show that π‘π‘ πœ–(π‘₯) is a neighborhood of each of its point. For this, let π‘¦βˆˆπ‘π‘ πœ–(π‘₯) and 𝑦≠π‘₯. Then π‘¦βˆˆπΆπ‘†(π‘₯)βˆ’{π‘₯}. By Lemma 3.1, π‘π‘ πœ–(π‘₯)βˆ’{π‘₯}β‰‘π‘πΈπœ–(π‘₯)∩(𝐢𝑆(π‘₯)βˆ’{π‘₯}), is open in 𝑀𝐸. Hence there exists a 𝛿-Euclidean neighborhood 𝑁𝐸𝛿(𝑦) of 𝑦 such that 𝑁𝐸𝛿(𝑦)βŠ†π‘π‘ πœ–(π‘₯)βˆ’{π‘₯}. This implies that 𝑁𝑠𝛿(𝑦)βŠ†π‘πΈπ›Ώ(𝑦)βŠ†π‘π‘ πœ–(π‘₯). Therefore, π‘π‘ πœ–(π‘₯) is a neighborhood of 𝑦. Since π‘π‘ πœ–(π‘₯) is a neighborhood of π‘₯, the result follows.

In the following proposition a subset of 𝑀 is obtained which is open in 𝑀𝑠 but not in 𝑀𝐸.

Lemma 3.3. Let 𝑀 be the 𝑛-dimensional Minkowski space and π‘₯βˆˆπ‘€. Then(i)𝐢𝑆(π‘₯) is not open in 𝑀𝐸,(ii)𝐢𝑆(π‘₯) is open in 𝑀𝑠.

Proof . (i)  We assert that π‘₯ is not an interior point of 𝐢𝑆(π‘₯) in 𝑀𝐸. To prove the assertion, consider the Euclidean neighbourhood π‘πΈπœ–(π‘₯) of radius πœ– containing π‘₯. Then it is easy to see that π‘πΈπœ–(π‘₯) is not contained in 𝐢𝑆(π‘₯). Since π‘₯βˆˆπΆπ‘†(π‘₯), the result follows.
(ii)  Let π‘¦βˆˆπΆπ‘†(π‘₯). Then either π‘¦βˆˆπΆπ‘†(π‘₯)βˆ’{π‘₯} or 𝑦=π‘₯. If π‘¦βˆˆπΆπ‘†(π‘₯)βˆ’{π‘₯}, then, by Lemma 3.1, there exists a 𝛿, such that 𝑁𝐸𝛿(𝑦)βŠ†πΆπ‘†(π‘₯)βˆ’{π‘₯} and hence 𝑁𝑠𝛿(𝑦)βŠ†πΆπ‘†(π‘₯). If 𝑦=π‘₯, then for any πœ–, π‘π‘ πœ–(π‘₯)βŠ†πΆπ‘†(π‘₯). Hence in either case, π‘₯ is an interior point of 𝐢𝑆(π‘₯) in 𝑀𝑠. This proves the result.

It is known that on 4-dimensional Minkowski space, 𝑠-topology is finer than the Euclidean topology. In the following proposition, we prove this result for the 𝑛-dimensional Minkowski space. In fact, the 𝑠-topology is shown to be strictly finer than the Euclidean topology.

Proposition 3.4. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then the 𝑠-topology on 𝑀 is strictly finer than the Euclidean topology on 𝑀.

Proof. Let 𝐺 be open in 𝑀𝐸 and π‘₯∈𝐺. Then there exists a Euclidean neighbourhoood π‘πΈπœ–(π‘₯) of π‘₯, such that π‘πΈπœ–(π‘₯)βŠ†πΊ. Hence π‘π‘ πœ–(π‘₯)βŠ†πΊ. This proves that 𝐺 is open in 𝑀𝑠. Hence the 𝑠-topology on 𝑀 is finer than the Euclidean topology on 𝑀. That it is strictly finer than the Euclidean topology follows from Lemma 3.3 (i) and (ii).

Lemma 3.5. (i)  Let 𝜎 be a spacelike straight line joining 𝑝 and π‘₯βˆˆπ‘€. Then π‘’βˆ’π‘£ is a spacelike vector for 𝑒,π‘£βˆˆπœŽ.
(ii)  Let 𝜏 be a timelike straight line joining 𝑝 and π‘₯βˆˆπ‘€. Then π‘’βˆ’π‘£ is a timelike vector for 𝑒,π‘£βˆˆπœ.
(iii)  Let πœ† be a light ray joining 𝑝 and π‘₯βˆˆπ‘€. Then π‘’βˆ’π‘£ is a lightlike vector for 𝑒,π‘£βˆˆπœ†.

Proof. (i)  For 𝑒,π‘£βˆˆπœŽ, there exist 𝛼,π›½βˆˆπ‘… such that 𝑒=𝑝+𝛼(π‘₯βˆ’π‘) and 𝑣=𝑝+𝛽(π‘₯βˆ’π‘). Then 𝑄(π‘’βˆ’π‘£)=(π›Όβˆ’π›½)2𝑄(π‘₯βˆ’π‘). This implies that π‘’βˆ’π‘£ is a spacelike vector, as 𝑄(π‘₯βˆ’π‘)<0.
(ii)  Similar to that of (i).
(iii)  Similar to that of (i).

Remark 3.6. Lemma 3.5 (i), (ii) and (iii) can be reinterpreted as follows.(i)If 𝜎 is a spacelike straight line, then for π‘€βˆˆπœŽ, 𝜎 is contained in 𝐢𝑆(𝑀).(ii)If 𝜏 is a timelike straight line, then for π‘€βˆˆπœ, 𝜏 is contained in 𝐢𝑇(𝑀).(iii)If πœ† is a light ray, then for π‘€βˆˆπœ†, πœ† is contained in 𝐢𝐿(𝑀).

Proposition 3.7. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then spacelike straight lines, timelike straight lines, and light rays are closed in 𝑀𝑠.

Proof. It follows from Proposition 3.4 and the facts that the spacelike straight lines, timelike straight lines, and light rays are all closed in 𝑀𝐸.

It is mentioned in [3] that the 𝑠-topology on the 4-dimensional Minkowski space induces Euclidean topology on every spacelike hyperplane. In the following proposition, it is proved that the 𝑠-topology on 𝑛-dimensional Minkowski space induces Euclidean topology on every spacelike straight line.

Proposition 3.8. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then the subspace topology on a spacelike straight line induced from the 𝑠-topology on 𝑀 is same as the subspace topology induced from the Euclidean topology.

Proof. Let 𝜎 be the spacelike straight line joining π‘₯ and 𝑦. In view of the fact that the Euclidean topology on 𝑀 is coarser than 𝑠-topology, it is sufficient to show that for πœ–>0, π‘π‘ πœ–(π‘₯)∩𝜎 is open in 𝜎𝐸, for all π‘₯βˆˆπ‘€. This easily follows by noting that π‘π‘ πœ–ξ‚»π‘(π‘₯)∩𝜎=πΈπœ–ξ€·π‘(π‘₯)∩𝜎ifπ‘₯βˆˆπœŽπΈπœ–ξ€Έ(π‘₯)βˆ’{π‘₯}∩𝜎ifπ‘₯βˆ‰πœŽ.(3.1)

It has been stated in [3] that the 𝑠-topology on the 4-dimensional Minkowski space induces discrete topology on a light ray. The following proposition generalizes this result to the 𝑛-dimensional Minkowski space.

Proposition 3.9. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then the 𝑠-topology on 𝑀 induces discrete topology on a light ray.

Proof. Let πœ† be a light ray and π‘βˆˆπœ†. Then from Remark 3.6 (iii), it follows that πœ†βŠ†πΆπΏ(𝑝). Hence, for πœ–>0, π‘π‘ πœ–(𝑝)βˆ©πœ†={𝑝}. This proves the result.

It has been stated in [3] that the 𝑠-topology on the 4-dimensional Minkowski space induces discrete topology on a timelike straight line. Following proposition generalizes this result to the 𝑛-dimensional Minkowski space.

Proposition 3.10. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then the 𝑠-topology on 𝑀 induces discrete topology on a timelike straight line.

Proof. Similar to that of Proposition 3.9.

4. Separability and Countability Axioms

In this section, it is proved that 𝑀𝑠 is a separable, first countable space that is not second countable.

Proposition 4.1. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then 𝑀𝑠 is separable.

Proof. Since 𝐾𝑛 is countable, it remains to show that 𝐾𝑛 is dense in 𝑀𝑠. Hence, it is sufficient to show that for π‘₯βˆˆπ‘€ and πœ–>0, π‘π‘ πœ–(π‘₯)βˆ©πΎπ‘›β‰ πœ™. If π‘₯βˆˆπΎπ‘›, then π‘π‘ πœ–(π‘₯)βˆ©πΎπ‘›β‰ πœ™. So let π‘₯βˆ‰πΎπ‘›. Then π‘π‘ πœ–(π‘₯)βˆ©πΎπ‘›=(π‘π‘ πœ–(π‘₯)βˆ’{π‘₯})βˆ©πΎπ‘›=π‘πΈπœ–(π‘₯)∩(𝐢𝑆(π‘₯)βˆ’{π‘₯})βˆ©πΎπ‘›. From Lemma 3.1, it follows that π‘πΈπœ–(π‘₯)∩(𝐢𝑆(π‘₯)βˆ’{π‘₯}) is open in 𝑀𝐸. Since 𝐾𝑛 is dense in 𝑀𝐸, π‘π‘ πœ–(π‘₯)βˆ©πΎπ‘›β‰ πœ™. This completes the proof.

The following lemma puts an upper bound on the cardinality of the set 𝐢(𝑀𝑠,𝑅).

Lemma 4.2. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then the cardinality of the set 𝐢(𝑀𝑠,𝑅) of all continuous real-valued functions on 𝑀𝑠 is at most equal to 2β„΅0.

Proof. From Proposition 4.1, 𝑀𝑠 is separable. Let 𝐷 be a countable subset of 𝑀𝑠. Then |𝐢(𝐷,𝑅)| is at most equal to (|𝑅|)|𝐷|=(2β„΅0)β„΅0=2β„΅0. Since two continuous maps are equal if they agree on a dense subset, hence |𝐢(𝑀𝑠,𝑅)| is at most equal to 2β„΅0. This completes the proof.

Proposition 4.3. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then 𝑀𝑠 is first countable.

Proof. Given π‘₯βˆˆπ‘€, the collection πœ‚(π‘₯)={π‘π‘ πœ–(π‘₯)βˆΆπœ–βˆˆπΎ} is a countable local base at π‘₯ for the 𝑠-topology on 𝑀. This shows that 𝑀𝑠 is first countable.

Proposition 4.4. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then 𝑀𝑠 is not second countable.

Proof. Let 𝑀𝑠 be second countable. Then since second countability is a hereditary property, it follows that a light ray is second countable. From Proposition 3.9, the induced topology on a light ray is discrete and hence it is not second countable, a contradiction.

5. Separation Axioms

In this section, besides studying other properties, it is proved that 𝑀𝑠 is a nonregular space.

It is known that 𝑀𝑠, for 𝑛=4, is 𝑇2 and hence 𝑇1 [3]. Indeed 𝑀𝑠 is 𝑇2 for all 𝑛. In the following proposition, we prove that 𝑀𝑠 is not regular.

Proposition 5.1. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then 𝑀𝑠 is not regular.

Proof. Let πœ† be a light ray passing through 0. Then by Propositions 3.4 and 3.9, πœ† is a closed discrete subspace of 𝑀𝑠. Hence πœ†βˆ’{0} is closed in 𝑀𝑠. We claim that πœ†βˆ’{0} and 0 cannot be separated by disjoint open sets. For this, let 𝐺1 and 𝐺2 be open sets in 𝑀𝑠 containing 0 and πœ†βˆ’{0}, respectively. Then for some πœ–>0, 0βˆˆπ‘π‘ πœ–(0)β‰‘π‘πΈπœ–(0)βˆ©πΆπ‘ (0)βŠ†πΊ1. Notice that π‘πΈπœ–(0)βˆ©πœ†β‰ {0}, for otherwise {0} would be open in 𝐿𝐸, a contradiction. Choose π‘₯βˆˆπ‘πΈπœ–(0)βˆ©πœ†,π‘₯β‰ 0. Then π‘₯βˆˆπœ†βˆ’{0} and hence there exists a 𝛿>0 such that π‘₯βˆˆπ‘π‘ π›Ώ(π‘₯)βŠ†πΊ2. Then it can be verified that π‘π‘ πœ–(0)βˆ©π‘π‘ π›Ώ(π‘₯)β‰ πœ™. Hence, 𝐺2∩𝐺1β‰ πœ™. This completes the proof.

Proposition 5.2. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then 𝑀𝑠 is not normal.

Proof. Let 𝑀𝑠 be normal. Then since 𝑀𝑠 is 𝑇1, 𝑀𝑠 is 𝑇4. The fact that a 𝑇4 space is regular implies that 𝑀𝑠 is regular, a contradiction to Proposition 5.1.

The following remark gives an alternate proof to the fact that 𝑀𝑠 is not normal.

Remark 5.3. Let 𝑀𝑠 be normal, πœ† a light ray, and π΄βŠ†πœ†. Then by Propositions 3.7 and 3.9, πœ† is a closed discrete subspace of 𝑀𝑠. Hence 𝐴 and πœ†βˆ’π΄ are closed in 𝑀𝑠. Since 𝑀𝑠 is normal, by Urysohn’s lemma, there exists a continuous map π‘“βˆΆπ‘€π‘ β†’π‘… such that 𝑓(𝐴)={0} and 𝑓(πœ†βˆ’π΄)={1}. This implies there would be at least as many real-valued continuous functions on 𝑀𝑠 as there are subsets of πœ†. Hence |𝐢(𝑀𝑠,𝑅)| would be at least (22β„΅0), a contradiction to Lemma 4.2.

Corollary 5.4. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then 𝑀𝑠 is not metrizable.

Proof. Since a metrizable space is regular, the result follows from Proposition 5.1.

6. Connectedness and Compactness

In this section, it is proved that 𝑀𝑠 is a path-connected, noncompact, non-LindelΓΆf, nonlocally compact, nonparacompact, non-locally π‘š-Euclidean space.

Proposition 6.1. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then 𝑀𝑠 is path-connected

Proof. Let π‘₯,π‘¦βˆˆπ‘€. Then either 𝑄(π‘₯βˆ’π‘¦)<0 or 𝑄(π‘₯βˆ’π‘¦)β‰₯0. If 𝑄(π‘₯βˆ’π‘¦)<0, define π›ΎβˆΆ[0,1]→𝑀𝑠 by 𝛾(𝑑)=π‘₯+𝑑(π‘¦βˆ’π‘₯). Then 𝛾(0)=π‘₯ and 𝛾(1)=𝑦. By Proposition 3.8, π›ΎβˆΆ[0,1]β†’[π‘₯,𝑦] is continuous. This implies that π›ΎβˆΆ[0,1]→𝑀𝑠 is continuous. Hence 𝛾 is the required path in 𝑀𝑠 joining π‘₯ and 𝑦. If 𝑄(π‘₯βˆ’π‘¦)β‰₯0, then choose π‘§βˆˆπΆπ‘ (π‘₯)βˆ©πΆπ‘ (𝑦). Define π›ΎβˆΆ[0,1]→𝑀𝑠 to be the join of 𝛾1∢[0,1]→𝑀𝑠 and 𝛾2∢[0,1]→𝑀𝑠, where 𝛾1[]𝛾(𝑑)=π‘₯+𝑑(π‘§βˆ’π‘₯);π‘‘βˆˆ0,12[].(𝑑)=π‘₯+𝑑(π‘¦βˆ’π‘§);π‘‘βˆˆ0,1(6.1)
Then by Proposition 3.8, 𝛾1∢[0,1]β†’[π‘₯,𝑧] and 𝛾2∢[0,1]β†’[𝑧,𝑦] are continuous. Hence 𝛾1∢[0,1]→𝑀𝑠 and 𝛾2∢[0,1]→𝑀𝑠 are continuous. Hence 𝛾1 and 𝛾2 are paths in 𝑀𝑠 joining π‘₯,𝑧 and 𝑧,𝑦, respectively. Since the join of two paths is a path, 𝛾 is the required path in 𝑀𝑠 joining π‘₯ and 𝑦. This completes the proof.

Corollary 6.2. 𝑀𝑠 is connected.

Proof. Since a path-connected space is connected, the result follows from Proposition 6.1.

It has been stated in [3] that the 4-dimensional Minkowski space with 𝑠-topology is not compact. The following proposition proves this result for 𝑛-dimensional Minkowski space.

Proposition 6.3. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then 𝑀𝑠 is not compact.

Proof. It follows from Proposition 3.4 and the fact that 𝑀𝐸 is not compact.

Proposition 6.4. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then 𝑀𝑠 is not LindelΓΆf.

Proof. Let 𝑀𝑠 be LindelΓΆf and πœ† a light ray. Then by Proposition 3.9, πœ† is a discrete subspace of 𝑀𝑠 and hence it is not LindelΓΆf. The fact that LindelΓΆfness is closed hereditary, together with Proposition 3.7, implies that πœ† is LindelΓΆf, a contradiction.

Proposition 6.5. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then 𝑀𝑠 is not paracompact.

Proof. Since a paracompact Hausdorff space is normal [9], hence 𝑀𝑠 is not paracompact from Proposition 5.2.

Proposition 6.6. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then 𝑀𝑠 is not locally compact.

Proof. Since a Hausdorff locally compact space is regular [9], the result follows from Proposition 5.1.

Proposition 6.7. Let 𝑀 be the 𝑛-dimensional Minkowski space. Then 𝑀𝑠 is not locally π‘š-Euclidean.

Proof. It follows from Proposition 6.6 and the fact that a locally π‘š-Euclidean space is locally compact [9].

7. Compact Sets and Simple Connectedness

The concept of Zeno sequences was originally defined by Zeeman [1] for 4-dimensional Minkowski space with fine topology. In this section, we develop the notion of Zeno sequence in 𝑛-dimensional Minkowski Space with 𝑠-topology to characterize the compact subsets of 𝑀𝑠. As a consequence of this study the two dimensional Minkowski space with 𝑠-topology is proved to be not simply connected. The study of Zeno sequences is also used to obtain a sufficient condition for continuity of maps from a topological space into 𝑀𝑠.

Definition 7.1. Let π‘§βˆˆπ‘€ and let (𝑧𝑛)π‘›βˆˆπ‘ be a sequence of distinct terms in 𝑀 such that 𝑧𝑛≠𝑧, for every π‘›βˆˆπ‘. Then (𝑧𝑛)π‘›βˆˆπ‘ is called a Zeno sequence in 𝑀𝑠 converging to π‘§βˆˆπ‘€, if (𝑧𝑛)π‘›βˆˆπ‘ converges to 𝑧 in 𝑀𝐸 but not in 𝑀𝑠. The image of a Zeno sequence (𝑧𝑛)π‘›βˆˆπ‘ will mean the set 𝑍={𝑧𝑛|π‘›βˆˆπ‘}. The completed image of a Zeno sequence (𝑧𝑛)π‘›βˆˆπ‘ will mean the set 𝑍βˆͺ{𝑧}.

Example 7.2. Let π‘§βˆˆπ‘€. Consider the collection {πœ†π‘›βˆΆπœ†π‘› is a light ray passing through 𝑧,π‘›βˆˆπ‘}. For π‘›βˆˆπ‘, choose π‘§π‘›βˆˆπœ†π‘› such that 0<𝑑𝐸(𝑧𝑛,𝑧)<1/𝑛 and 𝑧𝑛≠𝑧𝑖, for 𝑖=1,2,…,π‘›βˆ’1,𝑛>1. Then (𝑧𝑛)π‘›βˆˆπ‘ converges to 𝑧 in 𝑀𝐸 but not in 𝑀𝑠, since any 𝑠-neighborhood about 𝑧 contains no 𝑧𝑛. Hence (𝑧𝑛)π‘›βˆˆπ‘ is a Zeno sequence in 𝑀𝑠.

Example 7.3. Let π‘§βˆˆπ‘€. Consider the collection {πœπ‘›βˆΆπœπ‘› is a timelike straight line passing through 𝑧,π‘›βˆˆπ‘}. For π‘›βˆˆπ‘, choose π‘§π‘›βˆˆπœπ‘› such that 0<𝑑𝐸(𝑧𝑛,𝑧)<1/𝑛 and 𝑧𝑛≠𝑧𝑖, for 𝑖=1,2,…,π‘›βˆ’1,𝑛>1. Then (𝑧𝑛)π‘›βˆˆπ‘ converges to 𝑧 in 𝑀𝐸 but not in 𝑀𝑠, since any 𝑠-neighborhood about 𝑧 contains no 𝑧𝑛. Hence (𝑧𝑛)π‘›βˆˆπ‘ is a Zeno sequence in 𝑀𝑠.

Example 7.4. Let π‘§βˆˆπ‘€. Consider the collection {πœŽπ‘›βˆΆπœŽπ‘› is a spacelike straight line passing through 𝑧,π‘›βˆˆπ‘}. For π‘›βˆˆπ‘, choose π‘§π‘›βˆˆπœŽπ‘› such that 0<𝑑𝐸(𝑧𝑛,𝑧)<1/𝑛 and 𝑧𝑛≠𝑧𝑖, for 𝑖=1,2,…,π‘›βˆ’1,𝑛>1. Then (𝑧𝑛)π‘›βˆˆπ‘ converges to 𝑧 in 𝑀𝐸 and in 𝑀𝑠. Hence (𝑧𝑛)π‘›βˆˆπ‘ is a not a Zeno sequence in 𝑀𝑠.

Proposition 7.5. Let (𝑧𝑛)π‘›βˆˆπ‘ be a Zeno sequence in 𝑀𝑠 converging to 𝑧. Then (𝑧𝑛)π‘›βˆˆπ‘ admits a subsequence whose image is closed in 𝑀𝑠 but not in 𝑀𝐸.

Proof. Since (𝑧𝑛)π‘›βˆˆπ‘ does not converge to 𝑧 in 𝑀𝑠, there exists an open set π‘ˆ in 𝑀𝑠 containing 𝑧 that leaves outside infinitely many terms of the sequence (𝑧𝑛)π‘›βˆˆπ‘. Let (π‘§π‘›π‘˜)π‘˜βˆˆπ‘ be the subsequence of (𝑧𝑛)π‘›βˆˆπ‘ formed by these infinitely many terms and let 𝐴 be its image. Clearly, (π‘§π‘›π‘˜)π‘˜βˆˆπ‘ converges to 𝑧 in 𝑀𝐸. Since π‘§βˆ‰π΄, 𝐴 is not closed in 𝑀𝐸. To see that 𝐴 is closed in 𝑀𝑠, notice first that any point of 𝑀 other than 𝑧 is not a limit point of 𝐴 in 𝑀𝐸 and hence in 𝑀𝑠. Further, since π‘ˆβˆ©π΄ is empty, 𝑧 is not a limit point of 𝐴 in 𝑀𝑠. Thus 𝐴 has no limit point in 𝑀𝑠. This completes the proof.

In the following proposition, it is proved that a compact subset of 𝑀𝑠 cannot contain a Zeno sequence.

Proposition 7.6. Let 𝐢 be a subset of 𝑀 and 𝐢𝑠 be a compact. Then 𝐢 does not contain image of a Zeno sequence.

Proof. To the contrary, let (𝑧𝑛)π‘›βˆˆπ‘ be a Zeno sequence converging to 𝑧. Then from Proposition 7.5, (𝑧𝑛)π‘›βˆˆπ‘ admits a subsequence whose image, say 𝐴, is closed in 𝑀𝑠 but not in 𝑀𝐸. Then 𝐴 is compact in 𝑀𝑠. This implies that 𝐴 is compact 𝑀𝐸 and hence closed in 𝑀𝐸, a contradiction to Proposition 7.5.

Lemma 7.7. Let 𝐢 be a subset of 𝑀, such that 𝐢 does not contain the completed image of any Zeno sequence. Then for π‘βˆˆπΆ and every open set 𝐺𝑠𝑝 in 𝑀𝑠 containing 𝑝, there exists an open set 𝐺𝐸𝑝 containing 𝑝 of 𝑀𝐸 such that πΆβˆ©πΊπΈπ‘βŠ†πΆβˆ©πΊπ‘ π‘.

Proof. Suppose for some π‘βˆˆπΆ and an open set 𝐺𝑠𝑝 in 𝑀𝑠 containing 𝑝, there is no open set 𝐺𝐸𝑝 in 𝑀𝐸 such that πΆβˆ©πΊπΈπ‘βŠ‚πΆβˆ©πΊπ‘ π‘. For each π‘›βˆˆπ‘, choose π‘₯π‘›βˆˆπΆβˆ©π‘πΈ1/𝑛(𝑝) such that π‘₯π‘›βˆ‰πΆβˆ©πΊπ‘ π‘ and π‘₯𝑛≠π‘₯𝑖, for 𝑖=1,2,…,π‘›βˆ’1,𝑛>1. Then (π‘₯𝑛)π‘›βˆˆπ‘ is a Zeno sequence in 𝑀𝑠 converging to 𝑝, which is a contradiction since completed image of (π‘₯𝑛)π‘›βˆˆπ‘ is contained in 𝐢.

The following proposition determines a class of subsets 𝐢 of 𝑀 for which 𝐢𝑠 = 𝐢𝐸.

Proposition 7.8. Let 𝐢 be a subset of 𝑀, such that 𝐢 does not contain completed image of any Zeno sequence. Then 𝐢𝑠 = 𝐢𝐸.

Proof. From Lemma 7.7, it follows that the subspace Euclidean topology on 𝐢 is finer than the subspace 𝑠-topology on 𝐢. Proposition 3.4 now completes the proof.

The following proposition characterizes the compact subset of 𝑀𝑠.

Proposition 7.9. Let 𝐢 be subset of 𝑀 such that 𝐢 does not contain the completed image of any Zeno sequence. Then 𝐢𝐸 is compact if and only if 𝐢𝑠 is compact.

Proof. It follows from Proposition 7.8.

The following proposition characterizes the continuous maps from a topological space into 𝑀𝑠.

Proposition 7.10. Let 𝑋 be a topological space and π‘“βˆΆπ‘‹β†’π‘€πΈ a map such that 𝑓(𝑋) does not contain completed image of any Zeno sequence. Then π‘“βˆΆπ‘‹β†’π‘€πΈ is continuous iff π‘“βˆΆπ‘‹β†’π‘€π‘  is continuous.

Proof. Let π‘“βˆΆπ‘‹β†’π‘€π‘  be continuous. Then by Proposition 3.4, π‘“βˆΆπ‘‹β†’π‘€πΈ is continuous. Conversely, let π‘“βˆΆπ‘‹β†’π‘€πΈ is continuous. Then π‘“βˆΆπ‘‹β†’π‘“(𝑋)𝐸 is continuous and hence by Proposition 7.8β€‰β€‰π‘“βˆΆπ‘‹β†’π‘“(𝑋)𝑠 is continuous. This proves that π‘“βˆΆπ‘‹β†’π‘€π‘  is continuous.

Lemma 7.11. Let 𝐺 be an open set in 𝑀𝐸 and π‘§βˆˆπΊ. Then there exists a Zeno sequence in 𝑀𝑠 converging to 𝑧 with its terms in 𝐺.

Proof. Clearly π‘πΈπœ–(𝑧)βŠ†πΊ, for some πœ–>0. For π‘›βˆˆπ‘, choose π‘§π‘›βˆˆπ‘πΈπœ–(𝑧)∩𝐢𝐿(𝑧) such that 𝑑𝐸(𝑧𝑛,𝑧)<πœ–/𝑛 and 𝑧𝑛≠𝑧𝑖, for 𝑖=1,2,…,π‘›βˆ’1,𝑛>1. Then (𝑧𝑛)π‘›βˆˆπ‘ converges to 𝑧 in 𝑀𝐸 but not in 𝑀𝑠. This proves that (𝑧𝑛)π‘›βˆˆπ‘ is a Zeno sequence, as required.

Proposition 7.12. Let 𝑀 be the 2-dimensional Minkowski space. Then 𝑀𝑠 is not simply connected.

Proof. Since 𝑀𝑠 is path connected, it is sufficient to prove that the fundamental group of 𝑀𝑠 at some fixed base point is nontrivial. For this, fix the base point at (0, 0) denoted by 𝑂. Choose distinct ordered pairs of spacelike vectors (𝑒𝑖,𝑣𝑖) for 𝑖=1,2 such that π‘’π‘–βˆ’π‘£π‘– is a spacelike vector. For 𝑖=1,2, let π›Ύπ‘–βˆΆ[0,1]→𝑀𝑠 be defined by π›Ύπ‘ π‘–βŽ§βŽͺβŽͺ⎨βŽͺβŽͺ⎩(𝑑)=𝑂+3𝑑𝑒𝑖1;π‘ βˆˆ0,3𝑒𝑖+𝑣(3π‘‘βˆ’1)π‘–βˆ’π‘’π‘–ξ€Έξ‚ƒ1;π‘ βˆˆ3,23𝑣𝑖+(3π‘‘βˆ’2)π‘‚βˆ’π‘£π‘–ξ€Έξ‚ƒ1;π‘ βˆˆ3ξ‚„,1(7.1)
Then in view of Proposition 3.8 and the fact that the join of paths is a path, it follows that π›Ύπ‘–βˆΆ[0,1]→𝑀𝑠 is a path, for 𝑖=1,2. Since 𝛾𝑖(0)=𝛾𝑖(1), for 𝑖=1,2, hence 𝛾𝑖’s are loops based at 𝑂. We claim that 𝛾1 is not path homotopic to 𝛾2. Suppose, on the contrary, that they are path-homotopic. Let 𝐻∢[0,1]Γ—[0,1]→𝑀𝑠 be a path homotopy between 𝛾1 and 𝛾2 and 𝑇1,𝑇2 be the compact triangles in 𝑀𝐸 with boundaries 𝛾1([0,1]) and 𝛾2([0,1]), respectively. Then since (𝑒1,𝑣1)β‰ (𝑒2,𝑣2) at least one of int(𝑇1)βˆ’π‘‡2 or int(𝑇2)βˆ’π‘‡1 is nonempty, where int(𝐴) denotes the interior of set 𝐴 in 𝑀𝐸. Let int(𝑇1)βˆ’π‘‡2β‰ πœ™. If π‘βˆˆint(𝑇1)βˆ’π‘‡2, then π‘βˆˆπ»([0,1]Γ—[0,1]), for otherwise 𝐻 would be a path homotopy between 𝑇1 and 𝑇2 in the punctured plane π‘€πΈβˆ’{𝑝} which is not possible as 𝑇1 winds around 𝑝 while 𝑇2 does not. Hence int(𝑇1)βˆ’π‘‡2βŠ†π»([0,1]Γ—[0,1]). Since int(𝑇1)βˆ’π‘‡2 is open in 𝑀𝐸, from Lemma 7.11  int(𝑇1)βˆ’π‘‡2 contains image of a Zeno sequence in 𝑀𝑠 converging to 𝑝. This is a contradiction to Proposition 7.6, since 𝐻([0,1]Γ—[0,1]) is compact in 𝑀𝑠. This completes the proof.

8. Conclusion

The present paper is focused on a detailed topological study of the physically relevant 𝑠-topology on 4-dimensional Minkowski space. Often the mathematical structure of a physical theory, especially the topology on the underlying space, is never completely determined by the physics of the processes it seeks to describe. This nonuniqueness of the topology on underlying space motivates to identify and study those topologies that are significant from the perspective of the physical theory. One of the most important physical theories is the Einstein’s special theory of relativity, formulated on 4-dimensional Minkowski space, the underlying space for 𝑠-topology.

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