Lectures on Order and Topology Antonino Salibra 17 November 2014 1 Topology: main definitions and notation Definition 1.1 A topological space X is a pair X = (jXj; OX) where jXj is a nonempty set and OX is a family of subsets of jXj satisfying the following properties: (i) ;;X 2 OX; S (ii) If (Ui 2 OX : i 2 I) is a family of sets, then i2I Ui 2 OX; (iii) If U; V 2 OX then U \ V 2 OX. The elements of OX are called open sets.A closed set is the complement of an open set (i.e., A ⊆ jXj is closed iff jXj n A is open). The set of all closed sets is denoted by CX. The set CX of closed sets satisfies the following conditions: T (i) If (Ui 2 CX : i 2 I) is a family of closed sets, then i2I Ui 2 CX; (ii) If U; V 2 CX then U [ V 2 CX. A clopen set is a set which is both open and closed. The clopen subsets of a space constitutes a Boolean algebra. Example 1 (Euclidean topology) Let R be the set of real numbers. Given a < b 2 R, we denote by (a; b) = fx 2 R : a < x < bg the interval of all reals between a and b. The family of all sets U ⊆ R satisfying the following property (8x 2 U)(9a9b)(a < b) ^ x 2 (a; b) ^ (a; b) ⊆ U constitutes a topology on R. It is the usual euclidean topology of real line. The following are ex- amples of open sets: an interval (a; b); an infinite interval (a; +1); The following are examples of closed sets: a closed interval [a; b]; a closed infinite interval [a; +1). Example 2 (Discrete topology) Let X be a set. Then (X; P(X)), where P(X) is the set of all subsets of X, is a topological space. Every subset of X is open. This topology is called the discrete topology. Every subset of X is clopen. Example 3 (Indiscrete topology) Let X be a set. Then (X; f;;Xg) is a topological space. This topology is called the indiscrete topology. The only open sets are the empty set and the set X. 1 Example 4 (Cofinite topology) Let X be a set. A subset Y of X is cofinite if X n Y is finite. The family of all cofinite subsets of X constitutes the so-called cofinite topology of X. The usual euclidean topologies are defined by a metric. Definition 1.2 A metric on a set jXj is a map d : jXj × jXj ! R+ satisfying the following properties: (i) d(x; y) = 0 iff x = y; (ii) d(x; y) = d(y; x); (iii) d(x; y) ≤ d(x; z) + d(z; y). If x 2 jXj and r > 0, then B(x; r) = fy : d(x; y) < rg is the ball of center x and radius r. Example 5 (Metric topology) Let X = (jXj; d) be a metric space. The topology induced by the metric d on jXj is defined as follows: U ⊆ jXj is open iff, for every x 2 U there exists r > 0 such that B(x; r) ⊆ U: The euclidean topology on R of Example 1 is induced by the metric d(x; y) = jx − yj, where jx − yj is the absolute value of x − y. 2 p 2 2 The euclidean topology on R is induced by the metric d(x; y) = (x1 − y1) + (x2 − y2) , where x = (x1; x2) and y = (y1; y2). Example 6 (Metric on strings) Let A be a finite alphabet, A∗ be the set of finite strings of alphabet A and the empty string. We define a metric d on A∗. Given two distinct strings α; β 2 A∗ we define: d(α; β) = 2−k; where k is the number of states of the least automata distinguishing α and β. We recall that a deterministic automata is a tuple (Q; A; q0; F; δ), where Q is the finite set of states, A is the alphabet, q0 2 Q is the initial state, F ⊆ Q is the set of final states, and δ : Q × A ! Q is a map. The map δ can be extended by induction to a map δ∗ : Q × A∗ ! Q, where α 2 A∗ and a 2 A: δ∗(q; ) = q; δ∗(q; αa) = δ(δ∗(q; α); a): The strings α and β are not distinguished by the automa iff δ∗(q; α) = δ∗(q; β) for every q 2 Q. Lemma 7 Let (jXj; d) be a metric space. Then for all x 6= y 2 jXj there exists r > 0 such that B(x; r) \ B(y; r) = ;. Proof. Define r = d(x; y)=4. The following is an example of topology which cannot be induced by a metric. Example 8 Let N be a the set of natural numbers. We denote by [n) = fx 2 N : x ≥ ng the set of naturals greater than or equal to n. Then (N; f;g[f[n): n 2 Ng) is a space, whose topology cannot be induced by a metric. The intersection of two nonempty open sets is always nonempty: [n) \ [k) = [max(k; n)) is infinite. For example, [5) \ [2) = [5). Then the conclusion follows from Lemma 7. 2 Example 9 (Alexandrov topology) Let X = (jXj; ≤) be a partially ordered set (poset, for short). A subset Y of jXj is an upper set if y 2 Y ^ x 2 jXj ^ (x ≥ y) ) x 2 Y: The family of all upper sets of X is a topology, called the Alexandrov topology. Example 10 (Sierpinski topology) Let 2 = f0; 1g. We can define two nontrivial topologies on 2: • The Sierpinski topology, whose open sets are: ;, f1g, f0; 1g. • The opposite of the Sierpinski topology, whose open sets are : ;, f0g, f0; 1g. The Sierpinski topology coincides with the Alexandrov topology if we order the set 2 as follows: 0 < 1. 1.1 Neighbourhood System Let X be a space. A set U is a neighbourhood of x (w.r.t. the topology OX) if there exists an open V such that x 2 V ⊆ U. We denote by NOx the set of all neighbourhoods of x. An open neighborhood of x is any neighbourhood U of x such that U is open. There is an alternative way to define a topology, by first defining a neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points. Definition 1.3 Let jXj be a set. A neighbourhood system on jXj is the assignment of a family Nx of subsets of jXj to each x 2 jXj, such that 1. x 2 U for every U 2 Nx; 2. U; V 2 Nx ) U \ V 2 Nx; 3. U 2 Nx ^ U ⊆ V ) V 2 Nx; 4. (8U 2 Nx)(9V 2 Nx) V ⊆ U ^ (8y 2 V ) V 2 Ny. Proposition 11 1. If X is a space, then the family Nx of neighbourhoods of x (w.r.t. the topology OX) constitutes a neighbourhood system on jXj. 2. If (Nx : x 2 X) is a neighbourhood system on jXj, then the family of sets U such that x 2 U ) U 2 Nx constitutes a topology on jXj. Both definitions are compatible: the topology obtained from the neighbourhood system defined using open sets of X is the original one OX, and vice versa when starting out from a neighbour- hood system. 3 1.2 Interior, Closure, adherent and limit point Let X be a space. 1. The interior A˚of a set A ⊆ jXj is the largest open subset of A: [ A˚ = fU : U 2 OX ^ U ⊆ Ag: We often write int(A) for A˚. Example 12 • In any space, the interior of the empty set is the empty set. • If X is the Euclidean space R of real numbers, then int[0; 1] = (0; 1). • If X is the Euclidean space R, then the interior of the set Q of rational numbers is empty. • If X is the complex plane R2, then int(f(x; y) 2 R2 : x2 + y2 ≤ 1g) = f(x; y) 2 R2 : x2 + y2 < 1g. • In the Euclidean space, the interior of any finite set is the empty set. • If one considers on R the discrete topology in which every set is open, then int([0; 1]) = [0; 1]. • If one considers on R the indiscrete topology in which the only open sets are the empty set and R itself, then int([0; 1]) is the empty set. 2. The closure A of a subset A of X is the intersection of all closed sets containing A: \ A = fU : U 2 CX ^ A ⊆ Ug: (fxg will be denoted by x). We sometimes write cl(A) for A. Proposition 13 A is the set of all points y such that U \A 6= ; for every open set U 2 Ny. Proof. If y 2 A n A, y 2 U open and U \ A = ;, then A ⊆ jXj n U. The set jXj n U is closed, contains A but it does not contains y. We get the contradiction y2 = A. Example 14 • Consider a sphere in 3-dimensions. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior (which is called an open 3-ball). It is useful to be able to distinguish between the interior of 3-ball and the surface, so we distinguish between the open 3-ball, and the closed 3-ball - the closure of the 3-ball.