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 = (|X|, OX) where |X| is a nonempty set and OX is a family of subsets of |X| satisfying the following properties: (i) ∅,X ∈ OX; S (ii) If (Ui ∈ OX : i ∈ I) is a family of sets, then i∈I Ui ∈ OX; (iii) If U, V ∈ OX then U ∩ V ∈ OX.

The elements of OX are called open sets.A closed set is the complement of an open set (i.e., A ⊆ |X| is closed iff |X| \ 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 ∈ CX : i ∈ I) is a family of closed sets, then i∈I Ui ∈ CX; (ii) If U, V ∈ CX then U ∪ V ∈ 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 ∈ R, we denote by (a, b) = {x ∈ R : a < x < b} the interval of all reals between a and b. The family of all sets U ⊆ R satisfying the following property (∀x ∈ U)(∃a∃b)(a < b) ∧ x ∈ (a, b) ∧ (a, b) ⊆ U constitutes a topology on R. It is the usual euclidean topology of real line. The following are examples of open sets: an interval (a, b); an inﬁnite interval (a, +∞); The following are examples of closed sets: a closed interval [a, b]; a closed inﬁnite interval [a, +∞).

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, {∅,X}) 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 \ 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 |X| is a map d : |X| × |X| → 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 ∈ |X| and r > 0, then B(x, r) = {y : d(x, y) < r} is the ball of center x and radius r.

Example 5 (Metric topology) Let X = (|X|, d) be a metric space. The topology induced by the metric d on |X| is deﬁned as follows:

U ⊆ |X| is open iff, for every x ∈ 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) = |x − y|, where |x − y| 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 ∈ 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 ∈ 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 ∈ A∗ and a ∈ 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 ∈ Q.

Lemma 7 Let (|X|, d) be a metric space. Then for all x 6= y ∈ |X| 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) = {x ∈ N : x ≥ n} the set of naturals greater than or equal to n. Then (N, {∅}∪{[n): n ∈ N}) 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 = (|X|, ≤) be a partially ordered set (poset, for short). A subset Y of |X| is an upper set if

y ∈ Y ∧ x ∈ |X| ∧ (x ≥ y) ⇒ x ∈ Y.

The family of all upper sets of X is a topology, called the Alexandrov topology.

Example 10 (Sierpinski topology) Let 2 = {0, 1}. We can deﬁne two nontrivial topologies on 2: • The Sierpinski topology, whose open sets are: ∅, {1}, {0, 1}.

• The opposite of the Sierpinski topology, whose open sets are : ∅, {0}, {0, 1}. 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 ∈ 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 |X| be a set. A neighbourhood system on |X| is the assignment of a family Nx of subsets of |X| to each x ∈ |X|, such that

1. x ∈ U for every U ∈ Nx;

2. U, V ∈ Nx ⇒ U ∩ V ∈ Nx;

3. U ∈ Nx ∧ U ⊆ V ⇒ V ∈ Nx;

4. (∀U ∈ Nx)(∃V ∈ Nx) V ⊆ U ∧ (∀y ∈ V ) V ∈ 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 |X|.

2. If (Nx : x ∈ X) is a neighbourhood system on |X|, then the family of sets U such that

x ∈ U ⇒ U ∈ Nx

constitutes a topology on |X|. 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 neighbourhood system.

3 1.2 Interior, Closure, adherent and limit point Let X be a space.

1. The interior A˚of a set A ⊆ |X| is the largest open subset of A: [ A˚ = {U : U ∈ OX ∧ U ⊆ A}.

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({(x, y) ∈ R2 : x2 + y2 ≤ 1}) = {(x, y) ∈ R2 : x2 + y2 < 1}. • 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 = {U : U ∈ CX ∧ A ⊆ U}.

({x} 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 ∈ Ny.

Proof. If y ∈ A \ A, y ∈ U open and U ∩ A = ∅, then A ⊆ |X| \ U. The set |X| \ U is closed, contains A but it does not contains y. We get the contradiction y∈ / 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. The closure of the open 3-ball is the open 3-ball plus the surface. • In any space, ∅ = cl(∅). • In any space X, |X| = |X|. • If X is the Euclidean space R of real numbers, then (0, 1) = [0, 1]. • If X is the Euclidean space R, then the closure of the set Q of rational numbers is the whole space R. We say that Q is dense in R.

4 • If X is the complex plane R2, then cl({z ∈ C : |z| > 1}) = {z ∈ C : |z| ≥ 1}. • If S is a ﬁnite subset of a Euclidean space, then S = S. • If one considers on R the discrete topology in which every set is closed (open), then (0, 1) = (0, 1). • If one considers on R the indiscrete topology in which the only closed (open) sets are the empty set and R itself, then (0, 1) = R. • In the Sierpinski space 2 (see Example 10) we have 0 = {0} and 1 = {0, 1}. These examples show that the closure of a set depends upon the topology of the underlying space. The closure of a set also depends upon in which space we are taking the closure. For example, if X is the set of rational numbers, with the usual relative topology induced by the Euclidean metric, and if S = {q ∈ Q : q2 > 2, q > 0}, then S is closed in Q; however, the closure√ of S in the Euclidean√ space R is the set of all real numbers greater than or equal to 2. Recall that 2 is irrational.

3. A point y is a adherent to A if (U ∩ A) 6= ∅ for every U ∈ Ny.

4. A point y is a limit point of A if (U ∩ A) \{y}= 6 ∅ for every U ∈ Ny. Every point in A \ A is a limit point. The closure of A is the union of A and its limit points. Let A = {(x, y): x2 + y2 < 1} ∪ {(2, 0)} in the plane with the Euclidean topology. Then A = {(x, y): x2 + y2 ≤ 1} ∪ {(2, 0)}, while the set of limit points is equal to {(x, y): x2 + y2 ≤ 1}.1

5. The boundary (or frontier) of A is the closed set A ∩ |X| \ A. For example, the frontier of {(x, y): x2 + y2 < 1} in the euclidean plane is the set {(x, y): x2 + y2 = 1}.

1.3 Bases Let X be a space. A family B of open sets is a base (subbase) if every open set OX is union of elements of B (is union of ﬁnite intersections of elements in B).

Example 15 The collection of all open intervals (a, b) (a, b ∈ R) in the real line forms a base for the euclidean topology, because (i) the intersection of any two open intervals is itself an open interval or empty; (ii) every open is union of intervals. The collection of all semi-inﬁnite intervals of the forms (−∞, a) and (a, +∞), where a is a real number, form a subbase. Any interval (a, b) = (a, +∞) ∩ (−∞, b).

A base is not unique. Many bases, even of different sizes, may generate the same topology. For example, the open intervals with rational endpoints are also a base for the euclidean real topology, as are the open intervals with irrational endpoints, but these two sets are completely disjoint and both properly contained in the base of all open intervals.

Deﬁnition 1.4 A family B of subsets of a set |X| is called a base if it satisﬁes the following two conditions: 1A point of A is isolated if the singleton of a is open in the subspace A.

5 1. For every x ∈ |X|, there is B ∈ B such that x ∈ B; 2. If A, B ∈ B and x ∈ A ∩ B, then there is C ∈ B such that x ∈ C ⊆ A ∩ B.

If B is a base on |X| then B generates the following topology: U is open iff there exists D ⊆ B such that U = S D. Any family B of subsets of a set |X|, such that T B = |X|, generates a topology on |X|. In fact, B generates the base constituted by the sets B1 ∩ · · · ∩ Bn (Bi ∈ B).

Example 16 Let Z be the set of all integers. We deﬁne a topology on Z to provide a topological proof that there exist inﬁnite prime numbers. For b > 0 and 0 ≤ a < b, let

b Na = {x ∈ Z : x ≡ a (mod b)} = {a, a ± b, a ± 2b, . . . , a ± kb, . . . }.

b In other words, Na is the set of all integers that are congruent to a modulo b. b • The sets Na (b > 0, 0 ≤ a < b) constitute a subbase and generate a base for a topology on . An element of the base is the set N b1 ∩ · · · ∩ N bn . This set is either empty or inﬁnite for Z a1 an the Chinese remainder theorem: x ≡ a1(mod b1); ... ; x ≡ an(mod bn) admits a solution iff ai ≡ aj mod gcd(b1, . . . , bn) for all i, j. If e is the least positive solution, then the set lcm(b1,...,bn) of all solution is Ne .

b • All elements of the base are clopen. The complement of Na is equal to

b b Z \ Na = ∪0≤i6=a

• Except for −1, +1 and 0, all integers have prime factors. Therefore each is contained in one or more N0,p, where p is prime. We thus arrive at the following identity, where P is the set of prime numbers: Z \ {−1, +1} = ∪p∈P N0,p. If the set P were finite, then the right hand side would be closed as the union of a finite number of closed sets. Then the set {−1, +1} would be open as a complement of a closed set. This would contradict that every open set is infinite.

References

[1] T. Lawson. Topology: A geometrical approach, Oxford University Press, 2003 [2] Gierz, Hofmann, Keimel, Lawson, Mislove, Scott. Continuous lattices and domains. Cam- bridge University Press, 2003.

[3] P. Johnstone. Stone Spaces. Cambridge University Press, 1982. [4] P. Odifreddi. Short Course on Logic, Algebra and Topology, 1995. [5] L.A. Steen and J.A. Seebach, Jr. Counterexamples in topology. Springer-Verlag. Second edi- tion, 1995