Notes on Topology. Andrew Forrester January 28, Notation 1. 2 The Big Picture 1

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1 Notes on Topology Andrew Forrester January 28, 2009 Contents 1 Notation 1 2 The Big Picture 1 3 Fundamental Concepts 2 4 Topological Spaces and Topologies Topological Spaces Examples of Topologies Comparing Topologies Comparing Topological Spaces Topological Objects and Techniques Sets Collections Functions on Topological Spaces Techniques Topological Properties Axioms? Further Concepts 6 8 Major Theorems 6 9 Algebraic Topology 6 1 Notation or = inclusive or ( A or B means A or B or both) xor = exclusive or ( A xor B means A or B but not both) (I would write ior and xor to clearly distinguish between the two, but I should train myself to be fluent with the common mathematical convention.) 2 The Big Picture Doughnuts and coffee cups... open sets, neighborhoods of points... connected and disconnected parts, paths and bridges... stretching and morphing 1

2 3 Fundamental Concepts A,, U set, (proper) sub/superset, empty set, universal set ( universe of discourse ) A collection (of sets) a objects, elements, members of sets a = b identical or indistinguishable, inclusion and containment (proper, ;, ) union ( OR), arbitrary unions A A A, (+) intersection ( AND), symmetric difference ( XOR), arbitrary intersections A A A Ā, A C complement ( absolute complement, relative to universal set) A B, A \ B difference (relative complement of B in A) A B = disjoint (empty intersection) DeMorgan s Laws P (A) power set (2 A ) A B cartesian product P Q antecedent (hypothesis), consequent (conclusion), vacuous truth P Q inverse Q P converse Q P contrapositive 4 Topological Spaces and Topologies 4.1 Topological Spaces Topology A topology on a set X is a collection T of subsets of X having the following properties: (1), X T (2) The union of the elements of any subcollection of T is in T. ( U τa U T where τ a T ; τ a is arbitrary ) (3) The intersection of the elements of any finite subcollection of T is in T. ( U τf U T where τ f T ; τ f is finite ) Topological Space The pair (X, T ) is a topological space, but sometimes X is called a topological space and T is not mentioned, if it is understood. Open Sets A subset U of X is called an open set of X (relative to T ) if it is a member of the collection T. 4.2 Examples of Topologies Trivial Topology (Indescrete Topology): T = {, X} Discrete Topology: T = P (X) or T = { {x} : x X } Inherited Topology (Subspace Topology): Order Topology: Product Topology: normal normal normal (or something like that) continue with this kind of information 2

3 4.3 Comparing Topologies Comparable: Two topologies T 1, T 2 are comparable if T 1 T 2 or T 1 T 2. (Then T 1 is comparable with T 2 and T 2 is comparable with T 1.) They are not comparable if each has an element that the other doesn t have. Finer: If T 1 T 2, then T 1 is finer than T 2 : 1 has more subsets of X than T 2. Coarser: If T 1 T 2, then T 1 is coarser than T 2. (Proper/strict containment or inclusion strictly finer or strictly coarser) 4.4 Comparing Topological Spaces Homeomorphic (Topologically Equivalent) 5 Topological Objects and Techniques Of what use are these concepts (basis, for example)? 5.1 Sets Interior Int A The interior of a set A is the union of all open sets in that set. Boundary(?) or Lining(?) or something? Closure Ā The closure of a set A is the intersection of all closed sets containing that set. Limit Point Closed Set... A set is closed iff it contains its limit points. Linear Continuum L A set L is called a linear continuum if it has more than one element, is simply ordered, and has the following properties: (1) the least upper bound property, (2) if x < y, z such that x < z < y. 5.2 Collections Basis B A basis (for a topology) on a set X is a collection B of subsets of X (called basis elements) such that (1) for each x X, there is at least one basis element B containing x (2) if x belongs to the intersection of two basis elements B 1 and B 2, then there is a basis element B 3 containing x such that B 3 B 1 B 2 3

4 T Generated by B If B is a basis for a topology on X, then we define the topology T generated by B as follows: A subset U of X is a member of T (it is open ) if for each x U, there is a basis element B B such that x B U. Subbasis S A subbasis S (for a topology) on X is a collection of subsets of X whose union equals X. T Generated by S The topology generated by the subbasis S is defined to be the collection T of all unions of finite intersections of elements of S. Cover(ing) C Given a topological space (X, T ) and a subspace Y, a cover of Y is a collection C of subsets of X such that Y C C C. C is an open cover if C T. Separation (U, V ) A separation of a topological space X is a pair U, V of nonempty disjoint open subsets of X whose union is X. For a topological subspace Y of X: A separation of Y is a pair of disjoint nonempty sets A and B whose union is Y, neither of which contains a limit point of the other. 5.3 Functions on Topological Spaces Continuous Function Open Function Path Given x, y X, a path in X from x to y is a continuous map f : [a, b] X of some closed interval in the real line into X, such that f(a) = x and f(b) = y. Homeomorphism (or Bicontinuous Function, Topological Mapping) Imbedding Projection π i 5.4 Techniques Constructing new spaces from existing ones Convergence 4

5 6 Topological Properties Topological properties are properties that are preserved under homeomorphism. (Make sure the following properties meet this definition.) (One can think of a homeomorphism as a transformation that deforms a mathematical object without tearing or folding.) Properties that don t fall under this category include size and straightness. Connectedness Connected A topological space X is said to be connected if a separation of X. Equivalently: A space X is connected iff the only clopen subsets of X are and X. Totally Disconnected A space is totally disconnected if its only connected subspaces are one-point sets. Path-Connected A space X is said to be path connected if every pair of points of X can be joined by a path in X. Locally Path Connected Compact A space X is locally path connected if it has a basis of path connected open sets. A topological space is called compact if every open cover of X has a finite subcover. Hausdorff Normal Regular First-Countable Second-Countable Lindelöf 6.1 Axioms? T 1 T 2 Theorems Connectedness Discrete topology Totally disconnected? Totally disconnected Discrete topology? The continuous image of a connected set is connected. (So a path connected space must be connected.) (See Munkres pg 155 for reference to converse counterexamples.) Compactness 5

6 Every closed subspace of a compact space is compact. Hausdorff-ness Every compact subspace of a Hausdorff space is closed. Property Relationships Normal Regular Hausdorff points are closed 7 Further Concepts Topological Metric Spaces Metrizable Spaces Fundamental Group, First Homotopy Group Theorems Every metrizable space is normal. 8 Major Theorems Intermediate Value Theorem Let f : X Y be a continuous map, where X is a connected space and Y is an ordered set in the order topology. If a and b are two points of X and if r Y lies between f(a) and f(b), then c X such that f(c) = r. Urysohn Lemma Look in the topology book. Also, there is a good sheet of theorems in my topology class notes (near the end). Take a look and add some (or all) of them here. (And what are T 1 and T 2?) 9 Algebraic Topology Homology Two objects are homologous if, together, they form a boundary for some region. Used in Green s theorem: certain important integrals over curves will have the same value for any two curves that are homologous. References [1] James R. Munkres: Topology: Second Edition, Pearson Education (2000) 6