Pdf on the concept of genus in topology and complex analysis. If j is a topological space homeomorphic to kwhere kis a the underlying space of a simplicial complex k. Definitions and the statement of the jordan theorem. Welcome to lecture 5, in our course analysis of a complex kind.
Complex functions and their mapping properties, their limits, continuity and differentiability, analytic functions, analytic branches of a multiplevalued function. Introduction to orbifolds april 25, 2011 1 introduction orbifolds lie at the intersection of many di erent areas of mathematics, including algebraic and di erential geometry, topology, algebra and string theory. A lebesgue measurable set a in the complex plane is said. Introduction to basic topology of the complex plane.
In the same way we extend complex topology to the transcomplex plane and establish some results about limits and continuity of transcomplex functions, analogous to complex functions 9. In order to study the behavior of such functions well need to first understand the basic objects involved, namely the complex numbers. The closure of your set is the entire complex plane, since your set consists of the complex plane. In 6 11 12 we introduce transreal calculus with a transreal topology that extends real topology. Mod01 lec03 topology of the complex plane parti youtube. On the concept of genus in topology and complex analysis. Arithmetic, geometry and topology the complex plane. The set of complex numbers in fact a field and topological vector space is denoted c \mathbfc or. The complex plane c inherits a norm and hence a metric space structure from r2. School of mathematics and statistics mt3503 complex analysis. We have started by making spaces using simplices as building blocks. Topology of the complex plane and holomorphic functions. Topology on a complex plane notes part iii topology on the.
Connectedness is one of the principal topological properties that are used to distinguish topological spaces a subset of a topological space x is a connected set if it is a connected space when viewed as a subspace of x. Complex analysis is the study of functions that live in the complex plane, i. Topology of the complex plane a consequence of the triangle inequality discussed in the previous lecture is that the function dz. A jordan arc in the plane is the image of an injective continuous map of a closed and bounded interval a, b into the plane. Mrq 2019 school of mathematics and statistics mt3503 complex analysis problem sheet i. An abstract manifold cameron krulewski, math 2 project i march 10, 2017 in this talk, we seek to generalize the concept of manifold and discuss abstract, or topological, manifolds. Complex numbers can be plotted on the complex plane. The sum of two complex numbers and multiplication by a real number. Complex analysis video course course outline complex numbers, the topology of the complex plane, the extended complex plane and its representation using the sphere. As such it has two vector space structures, one as a twodimensional vector space over. This is a collection of topology notes compiled by math 490 topology students at the university of michigan in the winter 2007 semester. Topology is a branch of geometry that studies the geometric objects, called topological spaces, under continuous maps.
We introduce the sun topology in the plane definition 1. Asidefromrnitself,theprecedingexamples are also compact. Many common spaces are topological spaces whose topology can be defined by a metric. For the rest of the course we will study functions on c the complex plane, focusing on those which satisfy the complex analogue of di.
It is not a triviality to study the topology of subsets of the plane r2. The approach adopted in this course makes plain the similarities between these different. Two topological spaces are considered the same if there is a con tinuous bijection between them. A first course in complex analysis sfsu math department san. Such spaces exhibit a hidden symmetry, which is the culminationof18. For the rest of the course we will study functions on c the complex plane, focusing on those which satisfy the complex analogue of di erentiability. Topology from the differentiable viewpoint by john w. The early 20th century saw the emergence of a number of theories whose power and utility reside in large part in their generality. Well examine the example of real projective space, and show that its a compact abstract manifold by realizing it as a quotient space. We will thus need the notions of convergence and limits which c possesses.
Complex analysis for mathematics and engineering, 3rd. We mainly concern ourselves with the complex plane c. Complex analysis is the culmination of a deep and farranging study of the fundamental notions of complex di. This book grew out of the authors notes for the complex analysis class which he taught during the spring quarter of 2007 and 2008. Well start by introducing the complex plane along with the algebra and geometry of complex numbers and make our way via differentiation, integration, complex dynamics and power series representation into territories at the edge of whats known today. On the concept of genus in topology and complex analysis article pdf available in notices of the american mathematical society 566 june 2009 with 292 reads how we measure reads. June 22, 2017 we propose a general topology optimization. One of the most energetic of these general theories was that of. The title of the book, topology of numbers, is intended to express this visual slant, where we are using the term topology with its general meaning of the spatial arrangement and interlinking of the components of a system. Given a z 0 2c, the open disc of radius raround z 0 is given by d rz 0 fz2c jjz z 0j jul, 2015 8. The real or complex projective plane and the projective plane of order 3 given above are examples of desarguesian projective planes. This onepoint compactification of the complex plane is very important in the theory of.
School of mathematics and statistics mt3503 complex. Clearly, k topology is ner than the usual topology. We use cookies to distinguish you from other users and to provide you with a better experience on our websites. But what if we have a space, and we want to break it up into simplices. Introductory topics of pointset and algebraic topology are covered in a series of. Nov 05, 20 mod01 lec03 topology of the complex plane parti nptelhrd. Topology is a branch of geometry that stud ies the geometric objects, called topological spaces, under continuous maps. The course covered elementary aspects of complex analysis such as the cauchy integral theorem, the residue theorem, laurent series, and the riemann mapping theorem with riemann surface theory. Note that the conjugate zof a point zis its mirror image in the real axis. In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Cas representing a point or a vector x,y in r2, and according to.
Use setbuilder notation to define, as well as the modulus of the difference z z0, which represents the distance between the points representing z and z0 in the complex plane. In fact it hints at the existance of simplicial complexes existing for all polytopes, e. Two topological spaces are considered the same if there is a continuous bijection between them. The projective planes that can not be constructed in this manner are called nondesarguesian planes, and the moulton plane given above is an example of one. In this section we will be specifically interested in topological properties of the complex plane. Not included in this book is the important but somewhat more sophisticated topic of spectral sequences. By drawing a picture in the complex plane, you should be able to come up with a geometric proof of this inequality. Topology is the study of shapes, and we need to name some shapes and find out how to write them down in complex notation, and learn about some concepts of topology in order to move to some really neat applications next week. Paulson school of engineering and applied sciences, harvard university, cambridge, ma 028 and 2department of electrical engineering, princeton university, princeton, nj, 08544 dated. Topology on a complex plane notes part iii topology on. For instance, the real line, the complex plane, and the cantor set can be thought of as the same set with different topologies. Find all di erent topologies up to a homeomorphism on a set consisting of 4 elements which make it a connected topological space. Today, well talk a little bit about topology in the plane.
This course provides an introduction to complex analysis, that is the theory of complex functions of a complex variable. Orbifolds were rst introduced into topology and di erential. B asic t opology t opology, sometimes referred to as othe mathematics of continuityo, or orubber sheet geometryo, or othe theory of abstract topo logical spaceso, is all of these, but, abo ve all, it is a langua ge, used by mathematicians in practically all branches of our science. The set c of complex numbers is naturally identified with the plane r2. Topologysimplicial complexes wikibooks, open books for an. This shows that the usual topology is not ner than k topology. Every closed set contains its closure the set in union with its boundary or frontier points. We will thus need the notions of convergence and limits which c possesses because it is a metric space in fact normed vector space. In the rest of the book, the calculus of complex numbers will be built on the properties that we develop in this. In this chapter, we will study the basic properties of the field of complex numbers. Since this is a textbook on algebraic topology, details involving pointset topology are often treated lightly or skipped entirely in the body of the text. Topologysimplicial complexes wikibooks, open books for.
For the proof it is first necessary to pass from the plane of complex numbers to a compact manifold. A concise course in complex analysis and riemann surfaces. So it is not surprising that much of what we know about the plane carries over to the complex numbers. Typically, they are marked by an attention to the set or space of all examples of a particular kind. How can one show that a set a of complex numbers is bounded if and only if, given z0. A more extreme example, discussed in topology texts. Can an entire, nonconstant function map the complex plane to a an open bounded set. Topology of the complex plane chapter 2 complex analysis. The pro nite topology on the group z of integers is the weakest topology. Topology in the complex plane andrew browder introduction. A metric space is a set x where we have a notion of distance. The extended plane and its spherical representation.
The closure of your set is the entire complex plane, since your set consists of the complex plane with the positive real axis removed. For instance, complex functions are necessarily analytic. Complex numbers and the topology of the complex plane 1. Mathematics 490 introduction to topology winter 2007 what is this. A jordan curve or a simple closed curve in the plane r2 is the image c of an injective continuous map of a circle into the plane, s1 r2. Note that there is no neighbourhood of 0 in the usual topology which is contained in 1.
Topology in the plane introduction to complex numbers. In mathematics the term topology is used to describe certain geometric properties of spaces. The triangulation of a polygon in the plane is a simplcial complex. Jan 17, 2008 every closed set contains its closure the set in union with its boundary or frontier points. Combinatorics and topology of line arrangements in the complex projective plane enrique artalbartolo communicated by ronald stern abstract. Part iii topology on the complex plane denition 9 neighbourhood of a complex number. We have introduced distances on c and c that turned. The topology it generates is known as the k topology on r. The complex numbers may be represented as points in the plane, with the real number 1 represented by the point 1. We use some results about betti numbers of coverings of complements of plane projective curves to discuss the problem of how combinatorics.
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