General Topology

Topology Revision Notes Peadar Coyle October 7, 2010 Abstract The purpose of these notes is to summarize the material of Section B of M338 Topology with the Open University. 1 Introduction These are my revision notes intertwined with questions for a Topology class. Particularly difficult examples are 4.1, et al 2 Surfaces Let us first consider the provisional definition of a Surface: Definition 2.1. A surface is a topological space (X,τ ) with the property that each point of X has a neighbourhood that is homeomorphic either to an open disc in the plane or to an open half-disc in the upper half-plane, i.e. given any point x ∈ X, there is an open set U containing x such that U is homeomorphic either to an open disc in ℜ2 . The open set U is called a disc-like neighbourhood or half-disc-like neighbourhood of x, as the case may be. It is worth stressing that the idea of points of a surface having disc-like neighbourhoods or half-disc-like neighbourhoods is crucial to the concept of a surface. Definition 2.2. Let S = (X,τ ) be a surface. A boundary point of S is a point of X for which every neighbourhood contains a half-disc-like neighbourhood. The boundary of S is the set of all boundary points of S. A surface with a boundary is a surface whose boundary is non-empty. A surface without boundary is a surface with no boundary points. Theorem 2.3. The boundary of a surface is composed of a finite number of arcs, each of which is homeomorphic to a circle or to a unit interval, or it is the empty set. N-fold toruses are examples of a surface without a boundary. Hollow tubing surfaces can be constructed from a connected graph! In which the edges are thickened. The boundary number β is one of the three characteristic numbers that can be used to classify compact surfaces. 1 2.1 Homeomorphic surfaces Aim: to classify surfaces up to homeomorphism. Recall that a homeomorphism between two topological spaces (such as surfaces in space) is a bijection with the property that both f and its inverse f −1 preserve open sets. Intuitively we can treat surfaces in space as if they are rubbersheets. Theorem 2.4. Two surfaces that are homeomorphic have the same boundary number Remarks 1. This theorem applies to all surfaces and not just surfaces in space. 2. This theorem tells us that the boundary number is a topological invariant for surfaces, i.e. a property that is invariant under homeomorphisms. 3. It follows from the theorem that two surfaces with different boundary numbers cannot be homeomorphic. It does not follow that two surfaces with the same boundary number are homeomorphics - for example, a sphere and a torus both have boundary number 0, but are not homeomorphic. Definition 2.5. A compact surface is a surface that can be obtained from a polygon (or a finite number of polygons) by identifying edges. Definition 2.6. A surface is a compact path-connected Hausdorff topological space (X,τ ) with the property that, given any point x ∈ X, there is an open set U containing x such that U is homeomorphic either to an open disc in ℜ2 with the Euclidean topology or to an open half-disc in the upper half plane with the subspace topology inherited from the Euclidean topology on ℜ2 . A surface in space is a surface (X,τX ) where X is a subset of ℜ3 and τX is the subspace topology on X inherited from the Euclidean topology τ on ℜ3 . 3 Connected topological spaces Like in Graph theory, we discuss our definition of connectedness, by specifying what we mean by a disconnected topological space. 3.1 Disconnections and Connectedness Our task can be simply stated, answer the following question: Theorem 3.1. Given a topological space (X, T ), what should we mean by saying that X is disconnected? Definition 3.2. Let (X, T ) be a topological space. A disconnection {U, V } of X is a pair of disjoint non-empty open subsets, U and V, with X = U ∪ V. The space (X, T ) is disconnected if X has a disconnection. It is connected if X has no disconnection. A set A ⊆ X is connected if (A, TA ) is connected. Otherwise it is disconnected. (X, TX ) 2 4 A3: Topological Spaces 4.1 Bases Exercise A3.13 - E.Book Show that B = (a, b) : a, b ∈ R − Q with a < b is a base for the Euclidean topology on R. Answers An important comment is to immediately recognize that R-Q is the irrational numbers. To confirm something is a base, one needs to confirm two conditions (B1) and (B2) which are stated on p18 of the handbook. (B1) Since each set in B is an open interval with irrational endpoints, each set in B is an open set in the Euclidean topology, and so (B1) holds for B. (B2) Now suppose that U is an open set in the Euclidean topology. We must show that U can be written as the union of a family of sets from B. For each a ∈ U , there is an r ¿ 0 such that (a-r,a+r)⊆ U. Since for any two real numbers x ¡ y there is an irrational number z with x¡ z ¡y, we can find irrational numbers pa , qa ∈ R − Q for which a − r < pa < a and a < qa < a + r. Hence a ∈ (pa , qa ) ⊆ (a − r, a + r) ⊆ U, and (pa , qa ) ∈ B. It follows that U = [ [ (pa , qa ) ⊆ U, a ⊆ a∈U a∈U and so U = [ (pa , qa ), a∈U which is a union of sets from B. Thus B satisfies (B2). Since (B1) and (B2) are satisfied, B is a base for the Euclidean topology on R. 4.2 Canonical Form Equation Let us consider an example of ’obtaining the canonical form’ of an edge equation. The edge equation is abcab−1 dec−1 d−1 f e−1 g = 1. As a occurs twice in the same direction, we know that the surface is non-orientable. We first apply UML to bring the two occurences of a together: abcab−1 dec−1 d−1 f e−1 g = 1 → aac−1 b−1 b−1 dec−1 d−1 f e−1 g = 1 5 Notes from Day School The day school was informative, and a lot of difficult concepts were clarified. One of the take home points was to always begin from the definitions. Abstract Mathematics like Topology, mostly involves using the definitions and taking them as far as they can go. Another important thing is to actually think in terms of the exam, afterall one of the aims of this is to do work that will satisfy exam requirements. Its extremely important to prime the TMAs and follow some sort of schedule as much as possible. Topology is hard - it takes a certain amount of hard focus. 3 • To show something isn’t compact you just have to show one counterexample i.e. one open cover that isn’t a finite subcover • To prove it is, we need to show all covers of A have a finite subcover. I speculated that there seems to be some analogy in the sequences C3 with the notions of Big Oh Notation introduced in Graphs 4. However this analogy may be stretching it a bit. • Watch out for questions involving Bases for Topological Spaces • When inserting vertices in algebraic form of an object abcdea−1 d−1 use different colours, and proceed systematically! • Some excellent points in the notes I took, worth mining through for TEQ answers. • You can compare the number of cross caps derived from examining the canonical form with intuitive approach. Topology provides multiple ways of being able to solve and understand Shape and Space. I understand something in Mathematics when I can narrate the steps out loud, answer some TEQ questions about the underlying concepts and annotate a correct solution. Things should be practiced so much that they are intuitive. I speculated that Metric Space Ideas help with Hamming Distance concepts. Past papers would be an excellent thing to order. Careful with diagrams - need to include these in my studies - can’t opt of it. 4