1. Topological Spaces We Start with the Abstract Definition of Topological
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
07. Homeomorphisms and Embeddings
07. Homeomorphisms and embeddings Homeomorphisms are the isomorphisms in the category of topological spaces and continuous functions. Definition. 1) A function f :(X; τ) ! (Y; σ) is called a homeomorphism between (X; τ) and (Y; σ) if f is bijective, continuous and the inverse function f −1 :(Y; σ) ! (X; τ) is continuous. 2) (X; τ) is called homeomorphic to (Y; σ) if there is a homeomorphism f : X ! Y . Remarks. 1) Being homeomorphic is apparently an equivalence relation on the class of all topological spaces. 2) It is a fundamental task in General Topology to decide whether two spaces are homeomorphic or not. 3) Homeomorphic spaces (X; τ) and (Y; σ) cannot be distinguished with respect to their topological structure because a homeomorphism f : X ! Y not only is a bijection between the elements of X and Y but also yields a bijection between the topologies via O 7! f(O). Hence a property whose definition relies only on set theoretic notions and the concept of an open set holds if and only if it holds in each homeomorphic space. Such properties are called topological properties. For example, ”first countable" is a topological property, whereas the notion of a bounded subset in R is not a topological property. R ! − t Example. The function f : ( 1; 1) with f(t) = 1+jtj is a −1 x homeomorphism, the inverse function is f (x) = 1−|xj . 1 − ! b−a a+b The function g :( 1; 1) (a; b) with g(x) = 2 x + 2 is a homeo- morphism. Therefore all open intervals in R are homeomorphic to each other and homeomorphic to R . -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
Topological Properties
CHAPTER 4 Topological properties 1. Connectedness • Definitions and examples • Basic properties • Connected components • Connected versus path connected, again 2. Compactness • Definition and first examples • Topological properties of compact spaces • Compactness of products, and compactness in Rn • Compactness and continuous functions • Embeddings of compact manifolds • Sequential compactness • More about the metric case 3. Local compactness and the one-point compactification • Local compactness • The one-point compactification 4. More exercises 63 64 4. TOPOLOGICAL PROPERTIES 1. Connectedness 1.1. Definitions and examples. Definition 4.1. We say that a topological space (X, T ) is connected if X cannot be written as the union of two disjoint non-empty opens U, V ⊂ X. We say that a topological space (X, T ) is path connected if for any x, y ∈ X, there exists a path γ connecting x and y, i.e. a continuous map γ : [0, 1] → X such that γ(0) = x, γ(1) = y. Given (X, T ), we say that a subset A ⊂ X is connected (or path connected) if A, together with the induced topology, is connected (path connected). As we shall soon see, path connectedness implies connectedness. This is good news since, unlike connectedness, path connectedness can be checked more directly (see the examples below). Example 4.2. (1) X = {0, 1} with the discrete topology is not connected. Indeed, U = {0}, V = {1} are disjoint non-empty opens (in X) whose union is X. (2) Similarly, X = [0, 1) ∪ [2, 3] is not connected (take U = [0, 1), V = [2, 3]). More generally, if X ⊂ R is connected, then X must be an interval. -
Topology of Differentiable Manifolds
TOPOLOGY OF DIFFERENTIABLE MANIFOLDS D. MART´INEZ TORRES Contents 1. Introduction 1 1.1. Topology 2 1.2. Manifolds 3 2. More definitions and basic results 5 2.1. Submanifold vs. embedding 7 2.2. The tangent bundle of a Cr-manifold, r ≥ 1. 7 2.3. Transversality and submanifolds 9 2.4. Topology with Cr-functions. 9 2.5. Manifolds with boundary 13 2.6. 1-dimensional manifolds 16 3. Function spaces 19 4. Approximations 27 5. Sard's theorem and transversality 32 5.1. Transversality 35 6. Tubular neighborhoods, homotopies and isotopies 36 6.1. Homotopies, isotopies and linearizations 38 6.2. Linearizations 39 7. Degree, intersection number and Euler characteristic 42 7.1. Orientations 42 7.2. The degree of a map 43 7.3. Intersection number and Euler characteristic 45 7.4. Vector fields 46 8. Isotopies and gluings and Morse theory 47 8.1. Gluings 48 8.2. Morse functions 49 8.3. More on k-handles and smoothings 57 9. 2 and 3 dimensional compact oriented manifolds 60 9.1. Compact, oriented surfaces 60 9.2. Compact, oriented three manifolds 64 9.3. Heegard decompositions 64 9.4. Lens spaces 65 9.5. Higher genus 66 10. Exercises 66 References 67 1. Introduction Let us say a few words about the two key concepts in the title of the course, topology and differentiable manifolds. 1 2 D. MART´INEZ TORRES 1.1. Topology. It studies topological spaces and continuous maps among them, i.e. the category TOP with objects topological spaces and arrows continuous maps. -
1.1.3 Reminder of Some Simple Topological Concepts Definition 1.1.17
1. Preliminaries The Hausdorffcriterion could be paraphrased by saying that smaller neigh- borhoods make larger topologies. This is a very intuitive theorem, because the smaller the neighbourhoods are the easier it is for a set to contain neigh- bourhoods of all its points and so the more open sets there will be. Proof. Suppose τ τ . Fixed any point x X,letU (x). Then, since U ⇒ ⊆ ∈ ∈B is a neighbourhood of x in (X,τ), there exists O τ s.t. x O U.But ∈ ∈ ⊆ O τ implies by our assumption that O τ ,soU is also a neighbourhood ∈ ∈ of x in (X,τ ). Hence, by Definition 1.1.10 for (x), there exists V (x) B ∈B s.t. V U. ⊆ Conversely, let W τ. Then for each x W ,since (x) is a base of ⇐ ∈ ∈ B neighbourhoods w.r.t. τ,thereexistsU (x) such that x U W . Hence, ∈B ∈ ⊆ by assumption, there exists V (x)s.t.x V U W .ThenW τ . ∈B ∈ ⊆ ⊆ ∈ 1.1.3 Reminder of some simple topological concepts Definition 1.1.17. Given a topological space (X,τ) and a subset S of X,the subset or induced topology on S is defined by τ := S U U τ . That is, S { ∩ | ∈ } a subset of S is open in the subset topology if and only if it is the intersection of S with an open set in (X,τ). Alternatively, we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusion map ι : S X is continuous. -
16. Compactness
16. Compactness 1 Motivation While metrizability is the analyst's favourite topological property, compactness is surely the topologist's favourite topological property. Metric spaces have many nice properties, like being first countable, very separative, and so on, but compact spaces facilitate easy proofs. They allow you to do all the proofs you wished you could do, but never could. The definition of compactness, which we will see shortly, is quite innocuous looking. What compactness does for us is allow us to turn infinite collections of open sets into finite collections of open sets that do essentially the same thing. Compact spaces can be very large, as we will see in the next section, but in a strong sense every compact space acts like a finite space. This behaviour allows us to do a lot of hands-on, constructive proofs in compact spaces. For example, we can often take maxima and minima where in a non-compact space we would have to take suprema and infima. We will be able to intersect \all the open sets" in certain situations and end up with an open set, because finitely many open sets capture all the information in the whole collection. We will specifically prove an important result from analysis called the Heine-Borel theorem n that characterizes the compact subsets of R . This result is so fundamental to early analysis courses that it is often given as the definition of compactness in that context. 2 Basic definitions and examples Compactness is defined in terms of open covers, which we have talked about before in the context of bases but which we formally define here. -
Topology Via Higher-Order Intuitionistic Logic
Topology via higher-order intuitionistic logic Working version of 18th March 2004 These evolving notes will eventually be used to write a paper Mart´ınEscard´o Abstract When excluded middle fails, one can define a non-trivial topology on the one-point set, provided one doesn’t require all unions of open sets to be open. Technically, one obtains a subset of the subobject classifier, known as a dom- inance, which is to be thought of as a “Sierpinski set” that behaves as the Sierpinski space in classical topology. This induces topologies on all sets, rendering all functions continuous. Because virtually all theorems of classical topology require excluded middle (and even choice), it would be useless to reduce other topological notions to the notion of open set in the usual way. So, for example, our synthetic definition of compactness for a set X says that, for any Sierpinski-valued predicate p on X, the truth value of the statement “for all x ∈ X, p(x)” lives in the Sierpinski set. Similarly, other topological notions are defined by logical statements. We show that the proposed synthetic notions interact in the expected way. Moreover, we show that they coincide with the usual notions in cer- tain classical topological models, and we look at their interpretation in some computational models. 1 Introduction For suitable topologies, computable functions are continuous, and semidecidable properties of their inputs/outputs are open, but the converses of these two state- ments fail. Moreover, although semidecidable sets are closed under the formation of finite intersections and recursively enumerable unions, they fail to form the open sets of a topology in a literal sense. -
Connected Fundamental Groups and Homotopy Contacts in Fibered Topological (C, R) Space
S S symmetry Article Connected Fundamental Groups and Homotopy Contacts in Fibered Topological (C, R) Space Susmit Bagchi Department of Aerospace and Software Engineering (Informatics), Gyeongsang National University, Jinju 660701, Korea; [email protected] Abstract: The algebraic as well as geometric topological constructions of manifold embeddings and homotopy offer interesting insights about spaces and symmetry. This paper proposes the construction of 2-quasinormed variants of locally dense p-normed 2-spheres within a non-uniformly scalable quasinormed topological (C, R) space. The fibered space is dense and the 2-spheres are equivalent to the category of 3-dimensional manifolds or three-manifolds with simply connected boundary surfaces. However, the disjoint and proper embeddings of covering three-manifolds within the convex subspaces generates separations of p-normed 2-spheres. The 2-quasinormed variants of p-normed 2-spheres are compact and path-connected varieties within the dense space. The path-connection is further extended by introducing the concept of bi-connectedness, preserving Urysohn separation of closed subspaces. The local fundamental groups are constructed from the discrete variety of path-homotopies, which are interior to the respective 2-spheres. The simple connected boundaries of p-normed 2-spheres generate finite and countable sets of homotopy contacts of the fundamental groups. Interestingly, a compact fibre can prepare a homotopy loop in the fundamental group within the fibered topological (C, R) space. It is shown that the holomorphic condition is a requirement in the topological (C, R) space to preserve a convex path-component. Citation: Bagchi, S. Connected However, the topological projections of p-normed 2-spheres on the disjoint holomorphic complex Fundamental Groups and Homotopy subspaces retain the path-connection property irrespective of the projective points on real subspace. -
Basis (Topology), Basis (Vector Space), Ck, , Change of Variables (Integration), Closed, Closed Form, Closure, Coboundary, Cobou
Index basis (topology), Ôç injective, ç basis (vector space), ç interior, Ôò isomorphic, ¥ ∞ C , Ôý, ÔÔ isomorphism, ç Ck, Ôý, ÔÔ change of variables (integration), ÔÔ kernel, ç closed, Ôò closed form, Þ linear, ç closure, Ôò linear combination, ç coboundary, Þ linearly independent, ç coboundary map, Þ nullspace, ç cochain complex, Þ cochain homotopic, open cover, Ôò cochain homotopy, open set, Ôý cochain map, Þ cocycle, Þ partial derivative, Ôý cohomology, Þ compact, Ôò quotient space, â component function, À range, ç continuous, Ôç rank-nullity theorem, coordinate function, Ôý relative topology, Ôò dierential, Þ resolution, ¥ dimension, ç resolution of the identity, À direct sum, second-countable, Ôç directional derivative, Ôý short exact sequence, discrete topology, Ôò smooth homotopy, dual basis, À span, ç dual space, À standard topology on Rn, Ôò exact form, Þ subspace, ç exact sequence, ¥ subspace topology, Ôò support, Ô¥ Hausdor, Ôç surjective, ç homotopic, symmetry of partial derivatives, ÔÔ homotopy, topological space, Ôò image, ç topology, Ôò induced topology, Ôò total derivative, ÔÔ Ô trivial topology, Ôò vector space, ç well-dened, â ò Glossary Linear Algebra Denition. A real vector space V is a set equipped with an addition operation , and a scalar (R) multiplication operation satisfying the usual identities. + Denition. A subset W ⋅ V is a subspace if W is a vector space under the same operations as V. Denition. A linear combination⊂ of a subset S V is a sum of the form a v, where each a , and only nitely many of the a are nonzero. v v R ⊂ v v∈S v∈S ⋅ ∈ { } Denition.Qe span of a subset S V is the subspace span S of linear combinations of S. -
Section 11.3. Countability and Separability
11.3. Countability and Separability 1 Section 11.3. Countability and Separability Note. In this brief section, we introduce sequences and their limits. This is applied to topological spaces with certain types of bases. Definition. A sequence {xn} in a topological space X converges to a point x ∈ X provided for each neighborhood U of x, there is an index N such that n ≥ N, then xn belongs to U. The point x is a limit of the sequence. Note. We can now elaborate on some of the results you see early in a senior level real analysis class. You prove, for example, that the limit of a sequence of real numbers (under the usual topology) is unique. In a topological space, this may not be the case. For example, under the trivial topology every sequence converges to every point (at the other extreme, under the discrete topology, the only convergent sequences are those which are eventually constant). For an ad- ditional example, see Bonus 1 in my notes for Analysis 1 (MATH 4217/5217) at: http://faculty.etsu.edu/gardnerr/4217/notes/3-1.pdf. In a Hausdorff space (as in a metric space), points can be separated and limits of sequences are unique. Definition. A topological space (X, T ) is first countable provided there is a count- able base at each point. The space (X, T ) is second countable provided there is a countable base for the topology. 11.3. Countability and Separability 2 Example. Every metric space (X, ρ) is first countable since for all x ∈ X, the ∞ countable collection of open balls {B(x, 1/n)}n=1 is a base at x for the topology induced by the metric. -
Reviewworksheetsoln.Pdf
Math 351 - Elementary Topology Monday, October 8 Exam 1 Review Problems ⇤⇤ 1. Say U R is open if either it is finite or U = R. Why is this not a topology? ✓ 2. Let X = a, b . { } (a) If X is equipped with the trivial topology, which functions f : X R are continu- ! ous? What about functions g : R X? ! (b) If X is equipped with the topology ∆, a , X , which functions f : X R are { { } } ! continuous? What about functions g : R X? ! (c) If X is equipped with the discrete topology, which functions f : X R are contin- ! uous? What about functions g : R X? ! 3. Given an example of a topology on R (one we have discussed) that is not Hausdorff. 4. Show that if A X, then ∂A = ∆ if and only if A is both open and closed in X. ⇢ 5. Give an example of a space X and an open subset A such that Int(A) = A. 6 6. Let A X be a subspace. Show that C A is closed if and only if C = D A for some ⇢ ⇢ \ closed subset D X. ⇢ 7. Show that the addition function f : R2 R, defined by f (x, y)=x + y, is continuous. ! 8. Give an example of a function f : R R which is continuous only at 0 (in the usual ! topology). Hint: Define f piecewise, using the rationals and irrationals as the two pieces. Solutions. 1. This fails the union axiom. Any singleton set would be open. The set N is a union of single- ton sets and so should also be open, but it is not. -
Grothendieck-Topological Group Objects
GROTHENDIECK-TOPOLOGICAL GROUP OBJECTS JOAQU´IN LUNA-TORRES In memory of Carlos J. Ruiz Salguero Abstract. In analogy with the classical theory of topological groups, for finitely complete categories enriched with Grothendieck topologies, we provide the concepts of localized G-topological space, initial Grothendieck topologies and continuous morphisms, in order to obtain the concepts of G-topological monoid and G-topological group objects. 1. Introduction Topological groups (monoids) are very important mathematical objects with not only applications in mathematics, for example in Lie group theory, but also in physics. Topological groups are defined on topological spaces that involve points as fundamental objects of such topological spaces. The aim of this paper is to introduce a topological theory of groups within any category, finitely complete, enriched with Grothendieck topologies. The paper is organized as follows: We describe, in section 2, the notion of Grothendieck topology as in S. MacLane and I. Moerdijk [8] . In section 3, we present the concepts of localized G-topological space, initial Grothendieck topologies and continuous morphisms, and we study the lattice structure of all Grothendieck topologies on a category C ; after that, in section 4, as in M. Forrester-Baker [3], we present the concepts of monoid and group objects. In section 5 we present the concepts of G-topological monoid and G-topological group objects; finally, in section 6 we give some examples of arXiv:1909.11777v1 [math.CT] 25 Sep 2019 these ideas. 2. Theoretical Considerations Throughout this paper, we will work within an ambient category C which is finitely complete. From S. MacLane and I.