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A BRIEF OVERVIEW of ALEXANDROV SPACES Contents 1. Introduction 1 2. Preliminaries 1 3. Defining a Preorder on Alexandrov Spaces

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A BRIEF OVERVIEW of ALEXANDROV SPACES Contents 1. Introduction 1 2. Preliminaries 1 3. Defining a Preorder on Alexandrov Spaces A BRIEF OVERVIEW OF ALEXANDROV SPACES WILL ASNESS Abstract. This paper will discuss the Alexandrov spaces, an interesting kind of topological space that can be analyzed as a preorder. We will begin by defining what an Alexandrov space is, then discuss how we can analyze its topological properties as part of a preorder. Finally, we shall finish up with a quick look at some interesting topological properties of Alexandrov spaces that distinguish them from an arbitrary topological space. Contents 1. Introduction 1 2. Preliminaries 1 3. Defining a Preorder on Alexandrov Spaces 2 4. Separability 2 5. Analyzing Topology as a Preorder 3 6. Analyzing Paths within our Defined Preorder 8 Acknowledgments 10 References 10 1. Introduction In a discussion of an arbitrary topological space, an important axiom for a topology is that a finite inter- section of open sets must result in an open set- however, what if we discuss a topology such that an arbitrary intersection of open sets is open? Common examples of this kind of space are often far from the mind of the everyday topologist (think about the usual topology on R- it is not Alexandrov as the intersection of all open sets containing some point is the singleton set of that point, which is not open), and the more our discussion develops the more we shall find that these spaces do, indeed, maintain some very interesting properties. We call such spaces Alexandrov Spaces; soon, we shall show that we can view each Alexandrov space as a preorder and that, in fact, Alexandrov spaces and preorders are, essentially, the same. Through this lens, we find a very interesting way to analyze topology: as a preorder. We can even use this to find equivalences between ideas grounded in preorders and ideas in topology, and how an ordering-based concept can imply a topological concept (and vice versa). Simultaneously, we can find Alexandrov spaces to maintain some very interesting properties due to the laxity of the intersection condition of topologies (such as the equivalence of connectedness and path connectedness in these open spaces), which shall be discussed. Note that a thorough understanding of this paper will necessitate a solid understanding of the basics of order and topology. 2. Preliminaries In this section, we will briefly go over a few definitions and introduce some notations that shall ease our discussion of Alexandrov spaces (as well as make rigorous some definitions discussed above). Definition 2.1. A topological space X with a topology τ is an Alexandrov Space if, for any T ⊂ τ, \ T 2 τ: T 2T 1 Definition 2.2. In a space X with a topology τ, for all x 2 X let \ Ux = U: U open x2U Notice that, in a Alexandrov space, all Ux are open, as they are the intersection of open sets. This definition of Ux is vital to our discussion of Alexandrov spaces, as it becomes a very useful vehicle for analyzing the spaces. Throughout this paper, if A is used without being explicitly defined, regard it as some Alexandrov space. 3. Defining a Preorder on Alexandrov Spaces As mentioned above, we shall be showing that Alexandrov spaces and preorders are, essentially, equivalent. In the next two definitions, we define what a preorder is and then construct a preorder on any given Alexandrov space. Definition 3.1. In any set A, a binary relation ≤ is a preorder on A if ≤ is both reflexive and transitive. In other words, if x ≤ x for all x 2 A and, for all x; y; z 2 A such that x ≤ y and y ≤ z, we also have x ≤ z. Definition 3.2. In an Alexandrov space X, let ≤ be the binary relation such that if x1; x2 2 X, x1 ≤ x2 if Ux1 ⊂ Ux2 . Now that we have our binary relation defined, we demonstrate that it is a preorder. Proposition 3.3. The binary relation ≤ on an Alexandrov spaces X is a preorder. Proof. For all x 2 X, Ux ⊂ Ux, so x ≤ x and thus reflexivity is satisfied. Now, if x; y; z 2 X such that x ≤ y and y ≤ z, then Ux ⊂ Uy and Uy ⊂ Uz; as subset is a transitive relation, Ux ⊂ Uz, so x ≤ z, and thus transitivity of ≤ is maintained, so ≤ as defined is a preorder. Through the preceding proposition, we see that every Alexandrov space can be viewed as a preordered set; conversely, every preordered set can be viewed as an Alexandrov space. To do so, take any preordered set X and let B = ffx 2 X : x ≤ kg: k 2 Xg be a base for a topology τ on X (this is, in fact, an Alexandrov space!). Looking at our defined operations for defining a preorder on an Alexandrov space and constructing an Alexandrov space using a preorder, we see that these are inverse operations- so, we see our equivalence of Alexandrov Spaces and preorders! (In categorical language, we would see that we have an isomorphism of categories.) 4. Separability We begin a brief discussion of T0 and T1 topological spaces (known as two of the \separation axioms"), which will become important distinctions in some of our following results. Definition 4.1. A topological space X is T0 if, for all x1; x2 2 X, there exists an open set containing exactly one of the two points. Most see the T0 separation axiom as a relatively weak assumption of a space in a discussion of topology; essentially, in a T0 space, every point is \topologically distinct", by which we mean that, given two points in the space, there is at least one open set that contains one point and not the other, and so the topology has a way of seeing a difference between the two points. We now see how this idea of topological distinctiveness naturally leads into a discussion of partial orders. Definition 4.2. A binary relation ≤ on a set X is a partial order if it is a preorder and it satisfies the property: if x; y 2 X such that x ≤ y and y ≤ x, then x = y. This definition of a partial order says, essentially, that points can be distinguished by the partial order. Thus, with a clear similarity to the T0 separation axiom, we find the following useful result showing that a T0 Alexandrov space produces a partial order through our preorder described above. 2 Proposition 4.3. An Alexandrov space X is T0 if and only if ≤ is a partial order on X. Proof. Let x; y 2 X such that x 6= y. As X is T0, there exists some open set U ⊂ X such that x 2 U and y 62 U or y 2 U and x 62 U. Without loss of generality, x 2 U and y 62 U. As U is open, by the definition of Ux and the nature of set intersection, Ux ⊂ U. Therefore, as y 62 U, we have y 62 Ux; as y 2 Uy by defi- nition, we see that Ux 6⊂ Uy, so x 6≤ y. Therefore, we see that x 6= y implies that x 6≤ y or y 6≤ x. Thus, by contrapositive, x ≤ y and y ≤ x implies that x = y, and so we see that ≤ denotes a partial order in a T0 space. Now, for the other direction of the proof, assume that ≤ is a partial order. Now, fix x; y 2 X such that x 6= y. So, either x 6≤ y or y 6≤ x; without loss of generality, x 6≤ y. Thus, Ux 6⊂ Uy. Similarly to the previous paragraph, we can show that this implies that x 62 Uy; so, we see that X is T0, as y 2 Uy is open and x 62 Uy. We now delve into T1 spaces, and see why they can be rather uneventful in a discussion of Alexandrov spaces. Definition 4.4. A topological space X is T1 if, for all x1; x2 2 X, there exists an open set U1 ⊂ X such that x1 2 U1 and x2 62 U1 as well as an open set U2 ⊂ X such that x1 62 U2 and x2 2 U2. An intuition for T1 is that any point can be separated from another; essentially, any point can distance itself from another as it is in an open set not containing the other. The following proposition demonstrates why this property, combined with the Alexandrov property, can make a discussion of a topological space somewhat lacking. Proposition 4.5. If a topological space X is both T1 and Alexandrov, then the topology on it is discrete. Proof. Fix x 2 U. Now, fix some y 2 X such that y 6= x. As X is T1, there exists some open U ⊂ X such that x 2 U and y 62 U. By definition, Ux ⊂ U; thus, y 62 Ux. Therefore, as Ux is the intersection of all open sets containing x, we see that Ux = fxg. As X is Alexandrov, Ux is open; so, fxg is open. Thus, we see that all singleton sets are open, and therefore all subsets of X are open (as all sets are unions of singleton sets, which are all open). So, we see that the topology on X is the discrete topology. By Proposition 4:5, we see that it is rather trivial to discuss T1 Alexandrov spaces, as they are all the discrete topology, and thus no pair of points is comparable. Therefore, our discussion will (for the most part) avoid T1 spaces and center around T0 spaces. Although many of our theorems capture the general case, it is often useful to discuss only T0 spaces, which is not asking for too much as every space is homotopy equivalent to some T0 space.
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