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Math 535: Topology Homework 1
Math 535: Topology Homework 1

The θ-topology - some basic questions
The θ-topology - some basic questions

Final Exam on Math 114 (Set Theory)
Final Exam on Math 114 (Set Theory)

... Notes: The questions are not difficult as the number of points attached to them shows. Please read the text, understand the concepts and then read the questions attentively. Explain yourself clearly with short and correct English sentences. At most one idea per sentence! Use punctuation marks when n ...
Point-Set Topology: Glossary and Review.
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... is not Hausdorff a sequence can converge to more than one point.) continuous function. A function f : X → Y , where X and Y are topological spaces, is called continuous at p if for every open neighborhood U of f (p), the inverse image f −1 (U ) is open in X. If X and Y are metric spaces this reduces ...
Generalized Cohomology
Generalized Cohomology

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... Q Since (X, d) is a separable metric space, there exists aY countable dense set D = {yn }n∈N . Let Y = n∈N [0, 1], equipped with the product topology, denoted T . Consider the function φ:X→Y x 7→ φ(x) such that ∀n ∈ N, φn (x) = d(x, yn ) (by assumption on d, d(x, yn ) ∈ [0, 1]). Then we have • φ is ...
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ON b - δ - OPEN SETS IN TOPOLOGICAL SPACES

THE COARSE HAWAIIAN EARRING: A COUNTABLE SPACE WITH
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Available online through www.ijma.info ISSN 2229 – 5046
Available online through www.ijma.info ISSN 2229 – 5046

... (Ac)∪(Bc) is also rw-closed set in X. Therefore A∩B is rw-open set in X. Remark 2.11: The Union of two rw-open sets in X is need not be rw-open in X. Example 2.12: Let X={a,b,c,d} be a topological space with topology τ={φ,{a},{b},{a, b},{a, b, c}, X}. Then the set A={a,b}and B={d} are rw-open set in ...
Math 490 Extra Handout on the product topology and the box
Math 490 Extra Handout on the product topology and the box

Some results of semilocally simply connected property 1. Introduction
Some results of semilocally simply connected property 1. Introduction

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The Functor Category in Relation to the Model Theory of Modules

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... maps fi : X → Yi , i ∈ I, are continuous. In the light of this remark, we will call the topology T Φ the weak topology defined by Φ. Convergence can be nicely characterized: Proposition 3.5. Let X be a set, let Φ = (fi , Yi )i∈I be a family consisting of maps fi : X → Yi , where Yi is a topological ...
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... 1.3 Equivalent Formulations of Lebesgue Measurability The collection LRd of Lebesgue measurable subsets of Rd is closed under both countable unions and complements. Since LRd contains all of the open and closed subsets of Rd , it also contains all of the following types of sets. Definition 1.34. (a) ...
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Inner separation structures for topological spaces
Inner separation structures for topological spaces

... work, we will investigate the R-separated spaces, introduced in [3]. There, the author introduced the so called R-separation properties on a topological space X on which a binary relation R is defined. The main idea is to replace the identity relation in the classical separation properties with the ...
Homework Solutions 2
Homework Solutions 2

... from y on out to the boundary is  − d(y, x), which is positive. To show that B (x) is open, we must find an open ball about any y in this set which is entirely inside B (x). Let this ball be Bδ (y) where δ =  − d(y, x) > 0. If z is in this ball, then d(z, y) < δ = (−d(y, x)) and so d(z, x) ≤ d( ...
Aalborg Universitet A convenient category for directed homotopy Fajstrup, Lisbeth; Rosický, J.
Aalborg Universitet A convenient category for directed homotopy Fajstrup, Lisbeth; Rosický, J.

A convenient category - VBN
A convenient category - VBN

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Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They exist in several varieties such as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general cohomology theory, which encompasses also the ""usual"" topological cohomology theories such as singular cohomology. Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of D-modules, which provide applications to the theory of differential equations. In addition, generalisations of sheaves to more general settings than topological spaces, such as Grothendieck topology, have provided applications to mathematical logic and number theory.
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