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], -topology [3], and -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 -topology and -topology is a nonnormal, noncompact Hausdorff space besides proving that the homeomorphism group of 4-dimensional Minkowski space with space, and -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 -dimensional Minkowski space, , with the fine topology is separable, Hausdorff, nonnormal, nonlocally compact, non-Lindelöff 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 -topology besides studying its topological properties. It may be noted that -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 -topology on -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 -topology on -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 -topology. Topological properties of Minkowski space with -topology are dealt in Sections 4, 5, and 6. In Section 7, compact subsets of Minkowski space with -topology have been characterized. As a consequence of this study, it is proved that 2-dimensional Minkowski space with -topology is not simply connected. Finally, Section 8 concludes the paper.

2. Notation and Preliminaries

Let denote an indexing set while , , and denote the set of real, natural and rational numbers, respectively. To avoid any confusion later, we mention here that the symbol , in this paper, denotes the indefinite characteristic quadratic form. For a subset of a set , denotes the complement of in . For , let be the Euclidean distance between and . For denotes the -Euclidean neighborhood about given by the set . For , let denote the line segment joining and .

The -dimensional Minkowski Space, denoted by , is the -dimensional real vector space with a bilinear form , satisfying the following properties:(i)for all , , that is, the bilinear form is symmetric(ii)if for all , , then , that is, the bilinear form is nondegenerate, and(iii)there exists a basis for with The bilinear form is called the Lorentz inner product.

Elements of are referred to as events. If is an event, then the coordinate is called the time component and the coordinates are called the spatial components of relative to the basis . In terms of components, the Lorentz inner product of two events and is defined by . Lorentz inner product induces an indefinite characteristic quadratic form on given by . Thus . The group of all linear operators on which leave the quadratic form invariant, that is, , for all , is called the Lorentz group.

A event is called spacelike, lightlike (also called null) or timelike vector according as is negative, zero, or positive. The sets or = , or are likewise, respectively called the space cone, light cone (or null cone), and time cone at . For given , the set is called a spacelike straight line or light ray or timelike straight line joining and according as is negative or zero or positive. For further details, we refer to [8].

The Euclidean topology on the -dimensional Minkowski space is the topology generated by the basis . with the Euclidean topology will be denoted by .

The -topology on the -dimensional Minkowski space is defined by specifying the local base of neighborhoods at each point of given by the collection , where . We call the -neighborhood of radius . endowed with -topology is denoted by . For a subset of , () denotes the subspace of ().

3. Important Subsets and Subspaces of

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

Lemma 3.1. Let be the -dimensional Minkowski space and . Then , and are open in and is closed in .

Proof. For , define by . Then is continuous and and . Since and are open and is closed in , the results follow.

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

Lemma 3.2. Let be the -dimensional Minkowski space and . Then is open in .

Proof. It is sufficient to show that is a neighborhood of each of its point. For this, let and . Then . By Lemma 3.1, , is open in . Hence there exists a -Euclidean neighborhood of such that . This implies that . Therefore, is a neighborhood of . Since is a neighborhood of , the result follows.

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

Lemma 3.3. Let be the -dimensional Minkowski space and . Then(i) is not open in ,(ii) is open in .

Proof . (i)  We assert that is not an interior point of in . To prove the assertion, consider the Euclidean neighbourhood of radius containing . Then it is easy to see that is not contained in . Since , the result follows.
(ii)  Let . Then either or . If , then, by Lemma 3.1, there exists a , such that and hence . If , then for any , . Hence in either case, is an interior point of in . This proves the result.

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

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

Proof. Let be open in and . Then there exists a Euclidean neighbourhoood of , such that . Hence . This proves that is open in . Hence the -topology on is finer than the Euclidean topology on . That it is strictly finer than the Euclidean topology follows from Lemma 3.3 (i) and (ii).

Lemma 3.5. (i)  Let be a spacelike straight line joining and . Then is a spacelike vector for .
(ii)  Let be a timelike straight line joining and . Then is a timelike vector for .
(iii)  Let be a light ray joining and . Then is a lightlike vector for .

Proof. (i)  For , there exist such that and . Then . This implies that is a spacelike vector, as .
(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 is a spacelike straight line, then for , is contained in .(ii)If is a timelike straight line, then for , is contained in .(iii)If is a light ray, then for , is contained in .

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

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 .

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

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

Proof. Let be the spacelike straight line joining and . In view of the fact that the Euclidean topology on is coarser than -topology, it is sufficient to show that for , is open in , for all . This easily follows by noting that

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

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

Proof. Let be a light ray and . Then from Remark 3.6 (iii), it follows that . Hence, for , . This proves the result.

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

Proposition 3.10. Let be the -dimensional Minkowski space. Then the -topology on 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 is a separable, first countable space that is not second countable.

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

Proof. Since is countable, it remains to show that is dense in . Hence, it is sufficient to show that for and , . If , then . So let . Then . From Lemma 3.1, it follows that is open in . Since is dense in , . This completes the proof.

The following lemma puts an upper bound on the cardinality of the set .

Lemma 4.2. Let be the -dimensional Minkowski space. Then the cardinality of the set of all continuous real-valued functions on is at most equal to .

Proof. From Proposition 4.1, is separable. Let be a countable subset of . Then is at most equal to . Since two continuous maps are equal if they agree on a dense subset, hence is at most equal to . This completes the proof.

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

Proof. Given , the collection is a countable local base at for the -topology on . This shows that is first countable.

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

Proof. Let 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 is a nonregular space.

It is known that , for , is and hence [3]. Indeed is for all . In the following proposition, we prove that is not regular.

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

Proof. Let be a light ray passing through 0. Then by Propositions 3.4 and 3.9, is a closed discrete subspace of . Hence is closed in . We claim that and 0 cannot be separated by disjoint open sets. For this, let and be open sets in containing 0 and , respectively. Then for some , . Notice that , for otherwise would be open in , a contradiction. Choose . Then and hence there exists a such that . Then it can be verified that . Hence, . This completes the proof.

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

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

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

Remark 5.3. Let be normal, a light ray, and . Then by Propositions 3.7 and 3.9, is a closed discrete subspace of . Hence and are closed in . Since is normal, by Urysohn’s lemma, there exists a continuous map such that and . This implies there would be at least as many real-valued continuous functions on as there are subsets of . Hence would be at least , a contradiction to Lemma 4.2.

Corollary 5.4. Let be the -dimensional Minkowski space. Then 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 is a path-connected, noncompact, non-Lindelöf, nonlocally compact, nonparacompact, non-locally -Euclidean space.

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

Proof. Let . Then either or . If , define by . Then and . By Proposition 3.8, is continuous. This implies that is continuous. Hence is the required path in joining and . If , then choose . Define to be the join of and , where
Then by Proposition 3.8, and are continuous. Hence and are continuous. Hence and are paths in joining and , respectively. Since the join of two paths is a path, is the required path in joining and . This completes the proof.

Corollary 6.2. 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 -topology is not compact. The following proposition proves this result for -dimensional Minkowski space.

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

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

Proposition 6.4. Let be the -dimensional Minkowski space. Then is not Lindelöf.

Proof. Let be Lindelöf and a light ray. Then by Proposition 3.9, is a discrete subspace of and hence it is not Lindelöf. The fact that Lindelöfness is closed hereditary, together with Proposition 3.7, implies that is Lindelöf, a contradiction.

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

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

Proposition 6.6. Let be the -dimensional Minkowski space. Then 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 be the -dimensional Minkowski space. Then is not locally -Euclidean.

Proof. It follows from Proposition 6.6 and the fact that a locally -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 -dimensional Minkowski Space with -topology to characterize the compact subsets of . As a consequence of this study the two dimensional Minkowski space with -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 .

Definition 7.1. Let and let be a sequence of distinct terms in such that , for every . Then is called a Zeno sequence in converging to , if converges to in but not in . The image of a Zeno sequence will mean the set . The completed image of a Zeno sequence will mean the set .

Example 7.2. Let . Consider the collection is a light ray passing through . For , choose such that and , for . Then converges to in but not in , since any -neighborhood about contains no . Hence is a Zeno sequence in .

Example 7.3. Let . Consider the collection is a timelike straight line passing through . For , choose such that and , for . Then converges to in but not in , since any -neighborhood about contains no . Hence is a Zeno sequence in .

Example 7.4. Let . Consider the collection is a spacelike straight line passing through . For , choose such that and , for . Then converges to in and in . Hence is a not a Zeno sequence in .

Proposition 7.5. Let be a Zeno sequence in converging to . Then admits a subsequence whose image is closed in but not in .

Proof. Since does not converge to in , there exists an open set in containing that leaves outside infinitely many terms of the sequence . Let be the subsequence of formed by these infinitely many terms and let be its image. Clearly, converges to in . Since , is not closed in . To see that is closed in , notice first that any point of other than is not a limit point of in and hence in . Further, since is empty, is not a limit point of in . Thus has no limit point in . This completes the proof.

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

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

Proof. To the contrary, let be a Zeno sequence converging to . Then from Proposition 7.5, admits a subsequence whose image, say , is closed in but not in . Then is compact in . This implies that is compact and hence closed in , a contradiction to Proposition 7.5.

Lemma 7.7. Let be a subset of , such that does not contain the completed image of any Zeno sequence. Then for and every open set in containing , there exists an open set containing of such that .

Proof. Suppose for some and an open set in containing , there is no open set in such that . For each , choose such that and , for . Then is a Zeno sequence in converging to , which is a contradiction since completed image of is contained in .

The following proposition determines a class of subsets of for which = .

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

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

The following proposition characterizes the compact subset of .

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

Proof. It follows from Proposition 7.8.

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

Proposition 7.10. Let be a topological space and a map such that does not contain completed image of any Zeno sequence. Then is continuous iff is continuous.

Proof. Let be continuous. Then by Proposition 3.4, is continuous. Conversely, let is continuous. Then is continuous and hence by Proposition 7.8   is continuous. This proves that is continuous.

Lemma 7.11. Let be an open set in and . Then there exists a Zeno sequence in converging to with its terms in .

Proof. Clearly , for some . For , choose such that and , for . Then converges to in but not in . This proves that is a Zeno sequence, as required.

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

Proof. Since is path connected, it is sufficient to prove that the fundamental group of at some fixed base point is nontrivial. For this, fix the base point at (0, 0) denoted by . Choose distinct ordered pairs of spacelike vectors for such that is a spacelike vector. For , let be defined by
Then in view of Proposition 3.8 and the fact that the join of paths is a path, it follows that is a path, for . Since , for , hence ’s are loops based at . We claim that is not path homotopic to . Suppose, on the contrary, that they are path-homotopic. Let be a path homotopy between and and be the compact triangles in with boundaries and , respectively. Then since at least one of or is nonempty, where denotes the interior of set in . Let . If , then , for otherwise would be a path homotopy between and in the punctured plane which is not possible as winds around while does not. Hence . Since is open in , from Lemma 7.11   contains image of a Zeno sequence in converging to . This is a contradiction to Proposition 7.6, since is compact in . This completes the proof.

8. Conclusion

The present paper is focused on a detailed topological study of the physically relevant -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 -topology.