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Math 5853 Homework Solutions 36. a Map F : X → Y Is Called an Open
Math 5853 homework solutions 36. A map f : X → Y is called an open map if it takes open sets to open sets, and is called a closed map if it takes closed sets to closed sets. For example, a continuous bijection is a homeomorphism if and only if it is a closed map and an open map. 1. Give examples of continuous maps from R to R that are open but not closed, closed but not open, and neither open nor closed. open but not closed: f(x) = ex is a homeomorphism onto its image (0, ∞) (with the logarithm function as its inverse). If U is open, then f(U) is open in (0, ∞), and since (0, ∞) is open in R, f(U) is open in R. Therefore f is open. However, f(R) = (0, ∞) is not closed, so f is not closed. closed but not open: Constant functions are one example. We will give an example that is also surjective. Let f(x) = 0 for −1 ≤ x ≤ 1, f(x) = x − 1 for x ≥ 1, and f(x) = x+1 for x ≤ −1 (f is continuous by “gluing together on a finite collection of closed sets”). f is not open, since (−1, 1) is open but f((−1, 1)) = {0} is not open. To show that f is closed, let C be any closed subset of R. For n ∈ Z, −1 −1 define In = [n, n + 1]. Now, f (In) = [n + 1, n + 2] if n ≥ 1, f (I0) = [−1, 2], −1 −1 f (I−1) = [−2, 1], and f (In) = [n − 1, n] if n ≤ −2. -
A Grothendieck Site Is a Small Category C Equipped with a Grothendieck Topology T
Contents 5 Grothendieck topologies 1 6 Exactness properties 10 7 Geometric morphisms 17 8 Points and Boolean localization 22 5 Grothendieck topologies A Grothendieck site is a small category C equipped with a Grothendieck topology T . A Grothendieck topology T consists of a collec- tion of subfunctors R ⊂ hom( ;U); U 2 C ; called covering sieves, such that the following hold: 1) (base change) If R ⊂ hom( ;U) is covering and f : V ! U is a morphism of C , then f −1(R) = fg : W ! V j f · g 2 Rg is covering for V. 2) (local character) Suppose R;R0 ⊂ hom( ;U) and R is covering. If f −1(R0) is covering for all f : V ! U in R, then R0 is covering. 3) hom( ;U) is covering for all U 2 C . 1 Typically, Grothendieck topologies arise from cov- ering families in sites C having pullbacks. Cover- ing families are sets of maps which generate cov- ering sieves. Suppose that C has pullbacks. A topology T on C consists of families of sets of morphisms ffa : Ua ! Ug; U 2 C ; called covering families, such that 1) Suppose fa : Ua ! U is a covering family and y : V ! U is a morphism of C . Then the set of all V ×U Ua ! V is a covering family for V. 2) Suppose ffa : Ua ! Vg is covering, and fga;b : Wa;b ! Uag is covering for all a. Then the set of composites ga;b fa Wa;b −−! Ua −! U is covering. 3) The singleton set f1 : U ! Ug is covering for each U 2 C . -
On Sheaf Theory
Lectures on Sheaf Theory by C.H. Dowker Tata Institute of Fundamental Research Bombay 1957 Lectures on Sheaf Theory by C.H. Dowker Notes by S.V. Adavi and N. Ramabhadran Tata Institute of Fundamental Research Bombay 1956 Contents 1 Lecture 1 1 2 Lecture 2 5 3 Lecture 3 9 4 Lecture 4 15 5 Lecture 5 21 6 Lecture 6 27 7 Lecture 7 31 8 Lecture 8 35 9 Lecture 9 41 10 Lecture 10 47 11 Lecture 11 55 12 Lecture 12 59 13 Lecture 13 65 14 Lecture 14 73 iii iv Contents 15 Lecture 15 81 16 Lecture 16 87 17 Lecture 17 93 18 Lecture 18 101 19 Lecture 19 107 20 Lecture 20 113 21 Lecture 21 123 22 Lecture 22 129 23 Lecture 23 135 24 Lecture 24 139 25 Lecture 25 143 26 Lecture 26 147 27 Lecture 27 155 28 Lecture 28 161 29 Lecture 29 167 30 Lecture 30 171 31 Lecture 31 177 32 Lecture 32 183 33 Lecture 33 189 Lecture 1 Sheaves. 1 onto Definition. A sheaf S = (S, τ, X) of abelian groups is a map π : S −−−→ X, where S and X are topological spaces, such that 1. π is a local homeomorphism, 2. for each x ∈ X, π−1(x) is an abelian group, 3. addition is continuous. That π is a local homeomorphism means that for each point p ∈ S , there is an open set G with p ∈ G such that π|G maps G homeomorphi- cally onto some open set π(G). -
LECTURE 1. Differentiable Manifolds, Differentiable Maps
LECTURE 1. Differentiable manifolds, differentiable maps Def: topological m-manifold. Locally euclidean, Hausdorff, second-countable space. At each point we have a local chart. (U; h), where h : U ! Rm is a topological embedding (homeomorphism onto its image) and h(U) is open in Rm. Def: differentiable structure. An atlas of class Cr (r ≥ 1 or r = 1 ) for a topological m-manifold M is a collection U of local charts (U; h) satisfying: (i) The domains U of the charts in U define an open cover of M; (ii) If two domains of charts (U; h); (V; k) in U overlap (U \V 6= ;) , then the transition map: k ◦ h−1 : h(U \ V ) ! k(U \ V ) is a Cr diffeomorphism of open sets in Rm. (iii) The atlas U is maximal for property (ii). Def: differentiable map between differentiable manifolds. f : M m ! N n (continuous) is differentiable at p if for any local charts (U; h); (V; k) at p, resp. f(p) with f(U) ⊂ V the map k ◦ f ◦ h−1 = F : h(U) ! k(V ) is m differentiable at x0 = h(p) (as a map from an open subset of R to an open subset of Rn.) f : M ! N (continuous) is of class Cr if for any charts (U; h); (V; k) on M resp. N with f(U) ⊂ V , the composition k ◦ f ◦ h−1 : h(U) ! k(V ) is of class Cr (between open subsets of Rm, resp. Rn.) f : M ! N of class Cr is an immersion if, for any local charts (U; h); (V; k) as above (for M, resp. -
When Is a Local Homeomorphism a Semicovering Map? in This Section, We Obtained Some Conditions Under Which a Local Homeomorphism Is a Semicovering Map
When Is a Local Homeomorphism a Semicovering Map? Majid Kowkabia, Behrooz Mashayekhya,∗, Hamid Torabia aDepartment of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran Abstract In this paper, by reviewing the concept of semicovering maps, we present some conditions under which a local homeomorphism becomes a semicovering map. We also obtain some conditions under which a local homeomorphism is a covering map. Keywords: local homeomorphism, fundamental group, covering map, semicovering map. 2010 MSC: 57M10, 57M12, 57M05 1. Introduction It is well-known that every covering map is a local homeomorphism. The converse seems an interesting question that when a local homeomorphism is a covering map (see [4], [8], [6].) Recently, Brazas [2, Definition 3.1] generalized the concept of covering map by the phrase “A semicovering map is a local homeomorphism with continuous lifting of paths and homotopies”. Note that a map p : Y → X has continuous lifting of paths if ρp : (ρY )y → (ρX)p(y) defined by ρp(α) = p ◦ α is a homeomorphism for all y ∈ Y, where (ρY )y = {α : [0, 1] → Y |α(0) = y}. Also, a map p : Y → X has continuous lifting of homotopies if Φp : (ΦY )y → (ΦX)p(y) defined by Φp(φ)= p ◦ φ is a homeomorphism for all y ∈ Y , where elements of (ΦY )y are endpoint preserving homotopies of paths starting at y. It is easy to see that any covering map is a semicovering. qtop The quasitopological fundamental group π1 (X, x) is the quotient space of the loop space Ω(X, x) equipped with the compact-open topology with respect to the function Ω(X, x) → π1(X, x) identifying path components (see [1]). -
On Λ-Perfect Maps
Turkish Journal of Mathematics Turk J Math (2017) 41: 1087 { 1091 http://journals.tubitak.gov.tr/math/ ⃝c TUB¨ ITAK_ Research Article doi:10.3906/mat-1512-86 On λ-perfect maps Mehrdad NAMDARI∗, Mohammad Ali SIAVOSHI Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran Received: 22.12.2015 • Accepted/Published Online: 20.10.2016 • Final Version: 25.07.2017 Abstract: λ-Perfect maps, a generalization of perfect maps (i.e. continuous closed maps with compact fibers) are presented. Using Pλ -spaces and the concept of λ-compactness some classical results regarding λ-perfect maps will be extended. In particular, we show that if the composition fg is a λ-perfect map where f; g are continuous maps with fg well-defined, then f; g are α-perfect and β -perfect, respectively, on appropriate spaces, where α, β ≤ λ. Key words: λ-Compact, λ-perfect, Pλ -space, Lindel¨ofnumber 1. Introduction A perfect map is a kind of continuous function between topological spaces. Perfect maps are weaker than homeomorphisms, but strong enough to preserve some topological properties such as local compactness that are not always preserved by continuous maps. Let X; Y be two topological spaces such that X is Hausdorff. A continuous map f : X ! Y is said to be a perfect map provided that f is closed and surjective and each fiber f −1(y) is compact in X . In [3], the authors study a generalization of compactness, i.e. λ-compactness. Motivated by this concept we are led to generalize the concept of perfect mapping. -
MATH 552 C. Homeomorphisms, Manifolds and Diffeomorphisms 2020-09 Bijections Let X and Y Be Sets, and Let U ⊂ X Be a Subset. A
MATH 552 C. Homeomorphisms, Manifolds and Diffeomorphisms 2021-09 Bijections Let X and Y be sets, and let U ⊆ X be a subset. A map (or function) f with domain U takes each element x 2 U to a unique element f(x) 2 Y . We write y = f(x); or x 7! f(x): Note that in the textbook and lecture notes, one writes f : X ! Y; even if the domain is a proper subset of X, and often the domain is not specified explicitly. The range (or image) of f is the subset f(U) = f y 2 Y : there exists x 2 U such that y = f(x) g: A map f : X ! Y is one-to-one (or is invertible, or is an injection) if for all x1, x2 in the domain, f(x1) = f(x2) implies x1 = x2 (equivalently, x1 6= x2 implies f(x1) 6= f(x2)). If f is one-to-one, then there is an inverse map (or inverse function) f −1 : Y ! X whose domain is the range f(U) of f. A map f : X ! Y is onto (or is a surjection) if f(U) = Y . We will say a map f : X ! Y is a bijection (or one-to-one correspondence) from X onto Y if its domain is all of X, and it is one-to-one and onto. If f is a bijection, then its inverse map f −1 : Y ! X is defined on all of Y . Homeomorphisms Let X and Y be topological spaces. Examples of topological spaces are: i) Rn; ii) a metric space; iii) a smooth manifold (see below); iv) an open subset of Rn or of a smooth manifold. -
A CATEGORICAL INTRODUCTION to SHEAVES Contents 1
A CATEGORICAL INTRODUCTION TO SHEAVES DAPING WENG Abstract. Sheaf is a very useful notion when defining and computing many different cohomology theories over topological spaces. There are several ways to build up sheaf theory with different axioms; however, some of the axioms are a little bit hard to remember. In this paper, we are going to present a \natural" approach from a categorical viewpoint, with some remarks of applications of sheaf theory at the end. Some familiarity with basic category notions is assumed for the readers. Contents 1. Motivation1 2. Definitions and Constructions2 2.1. Presheaf2 2.2. Sheaf 4 3. Sheafification5 3.1. Direct Limit and Stalks5 3.2. Sheafification in Action8 3.3. Sheafification as an Adjoint Functor 12 4. Exact Sequence 15 5. Induced Sheaf 18 5.1. Direct Image 18 5.2. Inverse Image 18 5.3. Adjunction 20 6. A Brief Introduction to Sheaf Cohomology 21 Conclusion and Acknowlegdment 23 References 23 1. Motivation In many occasions, we may be interested in algebraic structures defined over local neigh- borhoods. For example, a theory of cohomology of a topological space often concerns with sets of maps from a local neighborhood to some abelian groups, which possesses a natural Z-module struture. Another example is line bundles (either real or complex): since R or C are themselves rings, the set of sections over a local neighborhood forms an R or C-module. To analyze this local algebraic information, mathematians came up with the notion of sheaves, which accommodate local and global data in a natural way. However, there are many fashion of introducing sheaves; Tennison [2] and Bredon [1] have done it in two very different styles in their seperate books, though both of which bear the name \Sheaf Theory". -
Midterm Exam
Prof. Alexandru Suciu MATH 4565 TOPOLOGY Spring 2018 MIDTERM EXAM 1. Consider the following two topologies on R2: (i) The Zariski topology, T1, where a base of open sets is formed by the comple- ments of zero sets of polynomials in two variables. (ii) The topology T2, the weakest topology where all straight lines are closed sets. 2 2 Show that (R ; T1) and (R ; T2) are not homeomorphic. 2. Define a topology T on the closed interval X = [−1; 1] by declaring a set to be open if it either does not contain the point 0, or it does contain (−1; 1). (a) Verify that T is indeed a topology. (b) What are the limit points of X? (c) Is X a T0 space? (I.e., given a pair of distinct points in X, does at least one of them have a neighborhood not containing the other?) (d) Is X a T1 space? (I.e., given a pair of distinct points in X, does each one of them have a neighborhood not containing the other?) (e) Show that X is compact. (f) Show that X locally path-connected. 3. Prove or disprove the following: (a) If X and Y are path-connected, then X × Y is path-connected. (b) If A ⊂ X is path-connected, then A is path-connected. (c) If X is locally path-connected, and A ⊂ X, then A is locally path-connected. (d) If X is path-connected, and f : X ! Y is continuous, then f(X) is path- connected. (e) If X is locally path-connected, and f : X ! Y is continuous, then f(X) is locally path-connected. -
On Semicovering, Subsemicovering, and Subcovering Maps
Archive of SID Journal of Algebraic Systems Vol. 7, No. 2, (2020), pp 227-244 ON SEMICOVERING, SUBSEMICOVERING, AND SUBCOVERING MAPS M. KOWKABI , B. MASHAYEKHY AND H. TORABI∗ Abstract. In this paper, by reviewing the concept of subcover- ing and semicovering maps, we extend the notion of subcovering map to subsemicovering map. We present some necessary or suffi- cient conditions for a local homeomorphism to be a subsemicover- ing map. Moreover, we investigate the relationship between these conditions by some examples. Finally, we give a necessary and sufficient condition for a subsemicovering map to be semicovering. 1. Introduction and Motivation Steinberg [8, Section 4.2] defined a map p : X~ ! X of locally path connected spaces as a subcovering map if there exist a covering map p0 : Y~ ! X and a topological embedding i : X~ ! Y~ such that p0 ◦ i = p. He presented a necessary and sufficient condition for a local homeomorphism p : X~ ! X to be subcovering. More precisely, he proved that a continuous map p : X~ ! X of locally path connected and semilocally simply connected spaces is subcovering if and only if p : X~ ! X is a local homeomorphism and any path f in X~ with p ◦ f null homotopic (in X) is closed, that is, f(0) = f(1) (see [8, Theorem 4.6]). Brazas [2, Definition 3.1] extended the concept of covering mapto semicovering map. A semicovering map is a local homeomorphism with continuous lifting of paths and homotopies. We simplified the notion of MSC(2010): Primary: 57M10; Secondary: 57M12, 57M05. Keywords: Local homeomorphism, fundamental group, covering map, semicovering map subcovering map, subsemicovering map. -
On Fundamental Groups of Quotient Spaces ∗ Jack S
Topology and its Applications 159 (2012) 322–330 Contents lists available at SciVerse ScienceDirect Topology and its Applications www.elsevier.com/locate/topol On fundamental groups of quotient spaces ∗ Jack S. Calcut a, , Robert E. Gompf b,1,JohnD.McCarthyc a Department of Mathematics, Oberlin College, Oberlin, OH 44074, United States b Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, United States c Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, United States article info abstract Article history: In classical covering space theory, a covering map induces an injection of fundamental Received 7 March 2011 groups. This paper reveals a dual property for certain quotient maps having connected Received in revised form 22 September fibers, with applications to orbit spaces of vector fields and leaf spaces in general. 2011 © 2011 Elsevier B.V. All rights reserved. Accepted 26 September 2011 MSC: primary 54B15 secondary 57M05, 37C10, 55U10 Keywords: Vietoris mapping theorem Homotopy Fundamental group Quotient map Connected fiber Orbit space Hawaiian earring Semilocally simply-connected 1. Introduction Given a map f : X → Y of topological spaces, an important problem is to relate the homology of X with that of Y using properties of f . Vietoris pioneered the study of this problem with his mapping theorem. Theorem. (Vietoris [32, §III]) Let f : X → Y be a surjective map of compact metric spaces. If the (Vietoris) homology of each fiber −1 Hr ( f (y)) is trivial for 0 r n − 1, then the induced homomorphism f∗ : Hr (X) → Hr (Y ) is an isomorphism for r n − 1 and is surjective for r = n. -
Local Mapping Relations and Global Implicit Function Theoremsc)
LOCAL MAPPING RELATIONS AND GLOBAL IMPLICIT FUNCTION THEOREMSC) BY WERNER C. RHEINBOLDT Dedicated to Professor H. Goertler on his sixtieth birthday 1. Introduction. Consider the problem of solving a nonlinear equation (1) F(x,y) = z with respect to y for given x and z. If, for example, x, y, z are elements of some Banach space, then under appropriate conditions about F, the well-known implicit function theorem ensures the "local" solvability of (1). The question arises when such a local result leads to "global" existence theorems. Several authors, for example, Cesari [2], Ehrmann [5], Hildebrandt and Graves [8], and Levy [9], have obtained results along this line. But these results are closely tied to the classical implicit function theorem, and no general theory appears to exist which permits the deduction of global solvability results once a local one is known. The problem has considerable interest in numerical analysis. In fact, when an operator equation (2) Fy = x is to be solved iteratively for y, the process converges usually only in a neighborhood of a solution. If for some other operator F0 the equation F0y = x has been solved, then one may try to "connect" F and F0 by an operator homotopy H(t,y) such that 77(0, y) = F0y and H(l,y) = Fy, and to "move" along the solution "curve" y(t) of H(t,y)=x, O^i^l, from the known solution y(0) to the unknown y(l), for instance, by using a locally convergent iterative process. This is the so-called "continuation method".