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SAM III General Topology Lecture 1 Contents Ordered sets Order and duality Galois connections Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces SAM III General Topology Lecture 1 SAM — Seminar in Abstract Mathematics [Version 20130202] is created by Zurab Janelidze at Mathematics Division, Stellenbosch Univeristy SAM III General Topology Lecture 1 Contents Ordered sets 1 Ordered sets Order and duality Galois connections Order and duality Galois connections Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces 2 Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces Order and duality SAM III General Topology Lecture 1 Contents Ordered sets Order and duality Galois connections Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces Definition An order 6 on a set X is a binary relation on X (i.e. a subset of X × X ) which is reflexive, antisymmetric, and transitive. An ordered set is a set equipped with an order, i.e. it is a pair (X , 6) where X is a set and 6 is an order on X . Dual of an ordered set Verify that if (X , 6) is an ordered set, then the pair (X , >) where the relation > is defined by x > y ⇔ y 6 x, is also an ordered set. It is called the dual of the ordered set (X , 6). Show that the dual of the dual of an ordered set (X , 6) is the same ordered set (X , 6). Ordered set of subsets Show that (P(X ), ⊆) is an ordered set. Galois connections SAM III General Topology Lecture 1 Contents Definition A Galois connection from an ordered set (X , 6) to an ordered set (Y , 6) is a pair (f , g ) of maps Ordered sets Order and duality Galois connections X f g /Y which are order-preserving and satisfy Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces o f (x) 6 y ⇔ x 6 g (y ) for all x ∈ X and y ∈ Y . In a Galois connection (f , g ), the map f is said to be the left adjoint map, and g is said to be the right adjoint map. Dual Galois connection Show that if (f , g ) is a Galois connection from (X , 6) to (Y , 6), then (g , f ) is a Galois connection from (Y , >) to (X , >). Galois connections: basic properties SAM III General Topology Lecture 1 Contents Ordered sets Order and duality Galois connections Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces A characterization of Galois connections Show that a pair (f , g ) of order-preserving maps X o f g / Y is a Galois connection from an ordered set (X , 6) to an ordered set (Y , 6) if and only if x 6 g (f (x)) for all x ∈ X , and f (g (y )) 6 y for all y ∈ Y . Uniqueness of adjoints Prove that if (f , g ) and (f , g 0 ) are Galois connections from an ordered set (X , 6) to an ordered set (Y , 6), then g = g 0 . By duality, deduce a similar fact for two Galois connections (f , g ) and (f 0 , g ). Galois connections: examples SAM III General Topology Lecture 1 Contents Galois connections induced by functions Ordered sets Consider a function f : X → Y from a set X to a set Y . Show that it induces a Galois connection Order and duality Galois connections Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces f (−) P(X ) o f −1 / P(Y ) (−) from the ordered set (P(X ), ⊆) to the ordered set (P(Y ), ⊆), where f (A) = {f (a) | a ∈ A} for each A ∈ P(X ). Topology via open sets SAM III General Topology Lecture 1 Contents Ordered sets Order and duality Galois connections Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces Definition A topology τ on a set X is a set of subsets of X , τ ⊆ P(X ), having the following three properties: {∅, X } ⊆ τ ; ∀A,B [({A, B} ⊆ τ ) ⇒ (A ∩ B ∈ τ )]; S ∀C [(C ⊆ τ ) ⇒ ( C ∈ τ )]. A topological space is a set equipped with a topology, i.e. it is a pair (X , τ ) where X is a set and τ is a topology on X . Elements of the topology τ are called open sets of the topological space, while elements of X are called points of the topological space. Alexandroff spaces SAM III General Topology Lecture 1 Contents Ordered sets Order and duality Galois connections Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces Topology is a generalization of order Verify that for an ordered set (X , 6) the set τ = {A ∈ P(X ) | ∀x∈A ∀y ∈X [(x 6 y ) ⇒ (y ∈ A)]} is a topology on X . It is called the Alexandroff topology of (X , 6). Show that two different orders on the same set give rise to two different Alexandroff topologies. Alexandroff spaces An Alexandroff space is a topological space in which arbitrary intersection of open sets is open. Show that for any fixed set X , there is a bijection between Alexandroff spaces (X , τ ) and pairs (X , 6) where 6 is a reflexive and transitive relation. Such pairs are called preorders. The preorder corresponding to an Alexandroff space is called its specialization preorder. Deduce that Alexandroff topology is merely the topology in an Alexandroff space corresponding to an ordered set. Adjunctions for preorders Generalize the notion of a Galois connection from ordered sets to preorders, calling the resulting notion an adjunction between preorders. Open intervals SAM III General Topology Lecture 1 Definition Contents Ordered sets Order and duality Galois connections Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces A linearly ordered set is an ordered set (X , 6) such that for any x ∈ X and y ∈ Y either x 6 y or y 6 x. Order topology In an ordered set (X , 6), we write x < y when x 6 y and x 6= y . An open interval is a subset I ⊆ X satisfying any one of the following three conditions: I There exists x ∈ X such that I = {i ∈ X | i < x}. I There exists x ∈ X such that I = {i ∈ X | x < i}. I There exist x, y ∈ X such that I = {i ∈ X | x < i < y }. In a linearly ordered set, is the set of arbitrary unions of open intervals a topology? “Cutting” and “gluing” spaces SAM III General Topology Lecture 1 Contents Subspaces Show that for a topological space (X , τ ) and a subset Y ⊆ X , the set Ordered sets Order and duality Galois connections Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces σ = {A ∩ Y | A ∈ τ } is a topology on Y . The topological space (Y , σ) is called a subspace of (X , τ ). Quotient spaces Show that for a topological space (X , τ ) and a partition P of X , the set σ = {V ⊆ P | [ V ∈ τ} is a topology on P, called a quotient topology. The topological space (P, σ) is called a quotient space. Adding and multiplying spaces SAM III General Topology Lecture 1 Contents Ordered sets Order and duality Galois connections Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces Product of two spaces Consider two topological spaces (X , τ ) and (Y , σ). Show that the set consisting of arbitrary unions of sets of the form A × B, where A ∈ τ and B ∈ σ, is a topology on X × Y . It is called the Tychonoff topology on the product X × Y . The corresponding topological space is called the product of (X , τ ) and (Y , σ). Sum of two spaces Consider two topological spaces (X , τ ) and (Y , σ). Show that the set consisting of sets of disjoint unions A t B where A ∈ τ and B ∈ σ, is a topology on the disjoint union X t Y . The resulting topological space is called the sum or coproduct of topological spaces (X , τ ) and (Y , σ). Product and sum of a family of spaces Define the product and the sum of an arbitrary family (Xi , τi )i∈I of topological spaces. Homeomorphism of spaces SAM III General Topology Lecture 1 Definition A topological space (X , τ ) is said to be homeomorphic to a topological space (Y , σ) when there is a bijection f : X → Y such that for any A ⊆ X , Contents A∈τ Ordered sets ⇔ f (A) ∈ σ. Order and duality Galois connections Topological spaces Topology via open sets Alexandroff spaces Open intervals Constructing spaces Homeomorphism of spaces Homeomorphic classes Show that the relation “homeomorphic” is an equivalence relation on the class of all topological spaces, i.e. show that it is reflexive, symmetric and transitive. The corresponding equivalence classes of spaces are called homeomorphic classes of spaces. Classification of spaces with at most three points Describe all homeomorphic classes of topological spaces having at most three points.