# A Study of Non-Euclideans-Topology

Authors: Gunjan Agrawal; Sampada Shrivastava
Journal: ISRN Mathematical Physics
 (2012)

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## Abstract

The present paper focuses on the characterization of compact sets of Minkowski space with a non-Euclideans-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-Lindelöf.


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## Body

## 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], t-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 t-topology and s-topology is a nonnormal, noncompact Hausdorff space besides proving that the homeomorphism group of 4-dimensional Minkowski space with space, t 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-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 t-topology besides studying its topological properties. It may be noted that t-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 thes-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 Section2. 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

LetΛ 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, X-A denotes the complement of A in X. For x,y∈Rn, let dE(x,y) be the Euclidean distance between x and y. For ϵ>0,NϵE(x) denotes the ϵ-Euclidean neighborhood about x given by the set {y∈Rn:dE(x,y)<ϵ}. For x,y∈Rn, let [x,y] denote the line segment joining x and y.Then-dimensional Minkowski Space, denoted by M, is the n-dimensional real vector space Rn with a bilinear form g:Rn×Rn→R, satisfying the following properties:(i)
for allx,y∈Rn, g(x,y)=g(y,x), that is, the bilinear form is symmetric(ii)
if for ally∈Rn, g(x,y)=0, then x=0, that is, the bilinear form is nondegenerate, and(iii)
there exists a basis{e0,e1,…,en-1} for Rn with
(2.1)g(ei,ej)={1ifi=j=0-1ifi=j=1,2,…,n-10ifi≠j.The bilinear form g is called the Lorentz inner product.Elements ofM are referred to as events. If x≡∑i=0n-1‍xiei is an event, then the coordinate x0 is called the time component and the coordinates x1,…,xn-1 are called the spatial components of x relative to the basis {e0,e1,…,en-1}. In terms of components, the Lorentz inner product g(x,y) of two events x≡∑i=0n-1‍xiei and y≡∑i=0n-1‍yiei is defined by x0y0-∑i=1n-1‍xiyi. Lorentz inner product induces an indefinite characteristic quadratic form Q on M given by Q(x)=g(x,x). Thus Q(x)=x02-∑i=1n-1‍xi2. The group of all linear operators T on M which leave the quadratic form Q invariant, that is, Q(x)=Q(T(x)), for all x∈M, is called the Lorentz group.A eventx∈M is called spacelike, lightlike (also called null) or timelike vector according as Q(x) is negative, zero, or positive. The sets CS(x)={y∈M:y=x or Q(y-x)<0},CL(x) = {y∈M:Q(y-x)=0}, CT(x)={y∈M:y=x or Q(y-x)>0} are likewise, respectively called the space cone, light cone (or null cone), and time cone at x. For given x,y∈M, the set {x+t(y-x)|t∈R} is called a spacelike straight line or light ray or timelike straight line joining x and y according as Q(y-x) is negative or zero or positive. For further details, we refer to [8].The Euclidean topology on then-dimensional Minkowski space M is the topology generated by the basis B={NϵE(x):ϵ>0,x∈M}. M with the Euclidean topology will be denoted by ME.Thes-topology on the n-dimensional Minkowski space M is defined by specifying the local base of neighborhoods at each point of x∈M given by the collection 𝒩(x)={Nϵs(x):ϵ>0}, where Nϵs(x)=NϵE(x)∩CS(x). We call Nϵs(x) the s-neighborhood of radius ϵ. 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 ofMs

In this section, besides proving that thes-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.
LetM be the n-dimensional Minkowski space and x∈M. Then CT(x)-{x}, and CS(x)-{x} are open in ME and CL(x) is closed in ME.Proof.
Foru∈ME, define f:ME→R by f(u)=(u0-x0)2-∑i=1n-1‍(ui-xi)2. Then f is continuous and f-1(0,∞)=CT(x)-{x},f-1(-∞,0)=CS(x)-{x} and f-1{0}=CL(x). Since (0,∞) and (-∞,0) are open and {0} is closed in ME, the results follow.In the following lemma, it is proved that thes-neighborhoods are open in Ms.Lemma 3.2.
LetM be the n-dimensional Minkowski space and x∈M. Then Nϵs(x),ϵ>0 is open in Ms.Proof.
It is sufficient to show thatNϵs(x) is a neighborhood of each of its point. For this, let y∈Nϵs(x) and y≠x. Then y∈CS(x)-{x}. By Lemma 3.1, Nϵs(x)-{x}≡NϵE(x)∩(CS(x)-{x}), is open in ME. Hence there exists a δ-Euclidean neighborhood NδE(y) of y such that NδE(y)⊆Nϵs(x)-{x}. This implies that Nδs(y)⊆NδE(y)⊆Nϵs(x). Therefore, Nϵs(x) is a neighborhood of y. Since Nϵs(x) is a neighborhood of x, the result follows.In the following proposition a subset ofM is obtained which is open in Ms but not in ME.Lemma 3.3.
LetM be the n-dimensional Minkowski space and x∈M. Then(i)
CS(x) is not open in ME,(ii)
CS(x) is open in Ms.Proof .
(i)  We assert thatx is not an interior point of CS(x) in ME. To prove the assertion, consider the Euclidean neighbourhood NϵE(x) of radius ϵ containing x. Then it is easy to see that NϵE(x) is not contained in CS(x). Since x∈CS(x), the result follows.
(ii)  Lety∈CS(x). Then either y∈CS(x)-{x} or y=x. If y∈CS(x)-{x}, then, by Lemma 3.1, there exists a δ, such that NδE(y)⊆CS(x)-{x} and hence Nδs(y)⊆CS(x). If y=x, then for any ϵ, Nϵs(x)⊆CS(x). Hence in either case, x is an interior point of CS(x) 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.
LetM be the n-dimensional Minkowski space. Then the s-topology on M is strictly finer than the Euclidean topology on M.Proof.
LetG be open in ME and x∈G. Then there exists a Euclidean neighbourhoood NϵE(x) of x, such that NϵE(x)⊆G. Hence Nϵs(x)⊆G. 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)  Letσ be a spacelike straight line joining p and x∈M. Then u-v is a spacelike vector for u,v∈σ.
(ii)  Letτ be a timelike straight line joining p and x∈M. Then u-v is a timelike vector for u,v∈τ.
(iii)  Letλ be a light ray joining p and x∈M. Then u-v is a lightlike vector for u,v∈λ.Proof.
(i)  Foru,v∈σ, there exist α,β∈R such that u=p+α(x-p) and v=p+β(x-p). Then Q(u-v)=(α-β)2Q(x-p). This implies that u-v is a spacelike vector, as Q(x-p)<0.
(ii)  Similar to that of (i).
(iii)  Similar to that of (i).Remark 3.6.
Lemma3.5 (i), (ii) and (iii) can be reinterpreted as follows.(i)
Ifσ is a spacelike straight line, then for w∈σ, σ is contained in CS(w).(ii)
Ifτ is a timelike straight line, then for w∈τ, τ is contained in CT(w).(iii)
Ifλ is a light ray, then for w∈λ, λ is contained in CL(w).Proposition 3.7.
LetM be the n-dimensional Minkowski space. Then spacelike straight lines, timelike straight lines, and light rays are closed in Ms.Proof.
It follows from Proposition3.4 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.
LetM 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.
Letσ be the spacelike straight line joining x 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 ϵ>0, Nϵs(x)∩σ is open in σE, for all x∈M. This easily follows by noting that
(3.1)Nϵs(x)∩σ={NϵE(x)∩σifx∈σ(NϵE(x)-{x})∩σifx∉σ.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.
LetM be the n-dimensional Minkowski space. Then the s-topology on M induces discrete topology on a light ray.Proof.
Letλ be a light ray and p∈λ. Then from Remark 3.6 (iii), it follows that λ⊆CL(p). Hence, for ϵ>0, Nϵs(p)∩λ={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.
LetM 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 Proposition3.9.


## 4. Separability and Countability Axioms

In this section, it is proved thatMs is a separable, first countable space that is not second countable.Proposition 4.1.
LetM be the n-dimensional Minkowski space. Then Ms is separable.Proof.
SinceKn is countable, it remains to show that Kn is dense in Ms. Hence, it is sufficient to show that for x∈M and ϵ>0, Nϵs(x)∩Kn≠ϕ. If x∈Kn, then Nϵs(x)∩Kn≠ϕ. So let x∉Kn. Then Nϵs(x)∩Kn=(Nϵs(x)-{x})∩Kn=NϵE(x)∩(CS(x)-{x})∩Kn. From Lemma 3.1, it follows that NϵE(x)∩(CS(x)-{x}) is open in ME. Since Kn is dense in ME, Nϵs(x)∩Kn≠ϕ. This completes the proof.The following lemma puts an upper bound on the cardinality of the setC(Ms,R).Lemma 4.2.
LetM be the n-dimensional Minkowski space. Then the cardinality of the set C(Ms,R) of all continuous real-valued functions on Ms is at most equal to 2ℵ0.Proof.
From Proposition4.1, Ms is separable. Let D be a countable subset of Ms. Then |C(D,R)| is at most equal to (|R|)|D|=(2ℵ0)ℵ0=2ℵ0. Since two continuous maps are equal if they agree on a dense subset, hence |C(Ms,R)| is at most equal to 2ℵ0. This completes the proof.Proposition 4.3.
LetM be the n-dimensional Minkowski space. Then Ms is first countable.Proof.
Givenx∈M, the collection η(x)={Nϵs(x):ϵ∈K} is a countable local base at x for the s-topology on M. This shows that Ms is first countable.Proposition 4.4.
LetM be the n-dimensional Minkowski space. Then Ms is not second countable.Proof.
LetMs 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 thatMs is a nonregular space.It is known thatMs, 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.
LetM be the n-dimensional Minkowski space. Then Ms is not regular.Proof.
Letλ be a light ray passing through 0. Then by Propositions 3.4 and 3.9, λ is a closed discrete subspace of Ms. Hence λ-{0} is closed in Ms. We claim that λ-{0} and 0 cannot be separated by disjoint open sets. For this, let G1 and G2 be open sets in Ms containing 0 and λ-{0}, respectively. Then for some ϵ>0, 0∈Nϵs(0)≡NϵE(0)∩Cs(0)⊆G1. Notice that NϵE(0)∩λ≠{0}, for otherwise {0} would be open in LE, a contradiction. Choose x∈NϵE(0)∩λ,x≠0. Then x∈λ-{0} and hence there exists a δ>0 such that x∈Nδs(x)⊆G2. Then it can be verified that Nϵs(0)∩Nδs(x)≠ϕ. Hence, G2∩G1≠ϕ. This completes the proof.Proposition 5.2.
LetM be the n-dimensional Minkowski space. Then Ms is not normal.Proof.
LetMs 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 thatMs is not normal.Remark 5.3.
LetMs be normal, λ a light ray, and A⊆λ. Then by Propositions 3.7 and 3.9, λ is a closed discrete subspace of Ms. Hence A and λ-A are closed in Ms. Since Ms is normal, by Urysohn’s lemma, there exists a continuous map f:Ms→R such that f(A)={0} and f(λ-A)={1}. This implies there would be at least as many real-valued continuous functions on Ms as there are subsets of λ. Hence |C(Ms,R)| would be at least (22ℵ0), a contradiction to Lemma 4.2.Corollary 5.4.
LetM be the n-dimensional Minkowski space. Then Ms is not metrizable.Proof.
Since a metrizable space is regular, the result follows from Proposition5.1.


## 6. Connectedness and Compactness

In this section, it is proved thatMs is a path-connected, noncompact, non-Lindelöf, nonlocally compact, nonparacompact, non-locally m-Euclidean space.Proposition 6.1.
LetM be the n-dimensional Minkowski space. Then Ms is path-connectedProof.
Letx,y∈M. Then either Q(x-y)<0 or Q(x-y)≥0. If Q(x-y)<0, define γ:[0,1]→Ms by γ(t)=x+t(y-x). Then γ(0)=x and γ(1)=y. By Proposition 3.8, γ:[0,1]→[x,y] is continuous. This implies that γ:[0,1]→Ms is continuous. Hence γ is the required path in Ms joining x and y. If Q(x-y)≥0, then choose z∈Cs(x)∩Cs(y). Define γ:[0,1]→Ms to be the join of γ1:[0,1]→Ms and γ2:[0,1]→Ms, where
(6.1)γ1(t)=x+t(z-x);t∈[0,1]γ2(t)=x+t(y-z);t∈[0,1].
Then by Proposition3.8, γ1:[0,1]→[x,z] and γ2:[0,1]→[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 x,z and z,y, respectively. Since the join of two paths is a path, γ is the required path in Ms joining x and y. This completes the proof.Corollary 6.2.
M
s is connected.Proof.
Since a path-connected space is connected, the result follows from Proposition6.1.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.
LetM be the n-dimensional Minkowski space. Then Ms is not compact.Proof.
It follows from Proposition3.4 and the fact that ME is not compact.Proposition 6.4.
LetM be the n-dimensional Minkowski space. Then Ms is not Lindelöf.Proof.
LetMs be Lindelöf and λ a light ray. Then by Proposition 3.9, λ is a discrete subspace of Ms and hence it is not Lindelöf. The fact that Lindelöfness is closed hereditary, together with Proposition 3.7, implies that λ is Lindelöf, a contradiction.Proposition 6.5.
LetM 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.
LetM 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.
LetM be the n-dimensional Minkowski space. Then Ms is not locally m-Euclidean.Proof.
It follows from Proposition6.6 and the fact that a locally m-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.
Letz∈M and let (zn)n∈N be a sequence of distinct terms in M such that zn≠z, for every n∈N. Then (zn)n∈N is called a Zeno sequence in Ms converging to z∈M, if (zn)n∈N converges to z in ME but not in Ms. The image of a Zeno sequence (zn)n∈N will mean the set Z={zn|n∈N}. The completed image of a Zeno sequence (zn)n∈N will mean the set Z∪{z}.Example 7.2.
Letz∈M. Consider the collection {λn:λn is a light ray passing through z,n∈N}. For n∈N, choose zn∈λn such that 0<dE(zn,z)<1/n and zn≠zi, for i=1,2,…,n-1,n>1. Then (zn)n∈N converges to z in ME but not in Ms, since any s-neighborhood about z contains no zn. Hence (zn)n∈N is a Zeno sequence in Ms.Example 7.3.
Letz∈M. Consider the collection {τn:τn is a timelike straight line passing through z,n∈N}. For n∈N, choose zn∈τn such that 0<dE(zn,z)<1/n and zn≠zi, for i=1,2,…,n-1,n>1. Then (zn)n∈N converges to z in ME but not in Ms, since any s-neighborhood about z contains no zn. Hence (zn)n∈N is a Zeno sequence in Ms.Example 7.4.
Letz∈M. Consider the collection {σn:σn is a spacelike straight line passing through z,n∈N}. For n∈N, choose zn∈σn such that 0<dE(zn,z)<1/n and zn≠zi, for i=1,2,…,n-1,n>1. Then (zn)n∈N converges to z in ME and in Ms. Hence (zn)n∈N is a not a Zeno sequence in Ms.Proposition 7.5.
Let(zn)n∈N be a Zeno sequence in Ms converging to z. Then (zn)n∈N admits a subsequence whose image is closed in Ms but not in ME.Proof.
Since(zn)n∈N does not converge to z in Ms, there exists an open set U in Ms containing z that leaves outside infinitely many terms of the sequence (zn)n∈N. Let (znk)k∈N be the subsequence of (zn)n∈N formed by these infinitely many terms and let A be its image. Clearly, (znk)k∈N converges to z in ME. Since z∉A, 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 U∩A 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 ofMs cannot contain a Zeno sequence.Proposition 7.6.
LetC be a subset of M and Cs be a compact. Then C does not contain image of a Zeno sequence.Proof.
To the contrary, let(zn)n∈N be a Zeno sequence converging to z. Then from Proposition 7.5, (zn)n∈N 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.
LetC be a subset of M, such that C does not contain the completed image of any Zeno sequence. Then for p∈C and every open set Gps in Ms containing p, there exists an open set GpE containing p of ME such that C∩GpE⊆C∩Gps.Proof.
Suppose for somep∈C and an open set Gps in Ms containing p, there is no open set GpE in ME such that C∩GpE⊂C∩Gps. For each n∈N, choose xn∈C∩N1/nE(p) such that xn∉C∩Gps and xn≠xi, for i=1,2,…,n-1,n>1. Then (xn)n∈N is a Zeno sequence in Ms converging to p, which is a contradiction since completed image of (xn)n∈N is contained in C.The following proposition determines a class of subsetsC of M for which Cs = CE.Proposition 7.8.
LetC be a subset of M, such that C does not contain completed image of any Zeno sequence. Then Cs = CE.Proof.
From Lemma7.7, it follows that the subspace Euclidean topology on C is finer than the subspace s-topology on C. Proposition 3.4 now completes the proof.The following proposition characterizes the compact subset ofMs.Proposition 7.9.
LetC be subset of M such that C does not contain the completed image of any Zeno sequence. Then CE is compact if and only if Cs is compact.Proof.
It follows from Proposition7.8.The following proposition characterizes the continuous maps from a topological space intoMs.Proposition 7.10.
LetX be a topological space and f:X→ME a map such that f(X) does not contain completed image of any Zeno sequence. Then f:X→ME is continuous iff f:X→Ms is continuous.Proof.
Letf:X→Ms be continuous. Then by Proposition 3.4, f:X→ME is continuous. Conversely, let f:X→ME is continuous. Then f:X→f(X)E is continuous and hence by Proposition 7.8  f:X→f(X)s is continuous. This proves that f:X→Ms is continuous.Lemma 7.11.
LetG be an open set in ME and z∈G. Then there exists a Zeno sequence in Ms converging to z with its terms in G.Proof.
ClearlyNϵE(z)⊆G, for some ϵ>0. For n∈N, choose zn∈NϵE(z)∩CL(z) such that dE(zn,z)<ϵ/n and zn≠zi, for i=1,2,…,n-1,n>1. Then (zn)n∈N converges to z in ME but not in Ms. This proves that (zn)n∈N is a Zeno sequence, as required.Proposition 7.12.
LetM be the 2-dimensional Minkowski space. Then Ms is not simply connected.Proof.
SinceMs 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 (ui,vi) for i=1,2 such that ui-vi is a spacelike vector. For i=1,2, let γi:[0,1]→Ms be defined by
(7.1)γis(t)={O+3tui;s∈[0,13]ui+(3t-1)(vi-ui);s∈[13,23]vi+(3t-2)(O-vi);s∈[13,1]
Then in view of Proposition3.8 and the fact that the join of paths is a path, it follows that γi:[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]×[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 (u1,v1)≠(u2,v2) at least one of int(T1)-T2 or int(T2)-T1 is nonempty, where int(A) denotes the interior of set A in ME. Let int(T1)-T2≠ϕ. If p∈int(T1)-T2, then p∈H([0,1]×[0,1]), for otherwise H would be a path homotopy between T1 and T2 in the punctured plane ME-{p} which is not possible as T1 winds around p while T2 does not. Hence int(T1)-T2⊆H([0,1]×[0,1]). Since int(T1)-T2 is open in ME, from Lemma 7.11  int(T1)-T2 contains image of a Zeno sequence in Ms converging to p. This is a contradiction to Proposition 7.6, since H([0,1]×[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 relevants-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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