Магистратура
2019/2020
Алгебраическая топология
Лучший по критерию «Полезность курса для Вашей будущей карьеры»
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Статус:
Курс обязательный (Математика)
Направление:
01.04.01. Математика
Кто читает:
Кафедра фундаментальной математики
Когда читается:
1-й курс, 1, 2 модуль
Формат изучения:
без онлайн-курса
Преподаватели:
Починка Ольга Витальевна
Прогр. обучения:
Математика
Язык:
английский
Кредиты:
5
Контактные часы:
60
Course Syllabus
Abstract
The course covers mainly those sections of topology that are directly related to the concept of a fundamental group. At the same time, it gives a fairly complete picture of the most typical ideas and methods of algebraic topology.
Learning Objectives
- The learning objectives are to grasp the concept of the fundamental group readily and provide a good introduction to what algebraic topology is about.
Expected Learning Outcomes
- The student is guided in the varieties of groups, able to operate with the kernel, image and factor group
- The student is able to establish the continuity of specific mappings and build continuous mappings between topological spaces
- A student is able to induce a topology from the surrounding space
- The student knows various ways to define equivalence relationships. Able to establish the homeomorphism of quotient spaces to standard topological objects
- Умеет индуцировать топологию с одного множества на другие
- The student knows how to check the axioms of a topological space, is well-versed in the most famous spaces
- A student knows the criterion of an open set in a Cartesian product and the universal property of a product
- A student knows the equivalent definitions of a compact set, knows how to prove the invariance of compactness
- A student is able to establish the Hausdorff property of a topological space, is able to prove the invariance of Hausdorff property
- A student knows how to find connected components of a topological space, knows how to prove connected invariance
Course Contents
- Sets and groupsWe give some of the basic definitions and results of set theory and group theory that are used in the future
- Continuous functionsWe generalize the notion of continuous function to topological spaces
- Induced topologyLet S be a subset of the topological space X. We can give S a topology from that of X
- Background: metric spacesWe shall find out what this 'structure' is by looking at euclidean and metric spaces
- Quotient topology and groups acting on spacesWe essentially considered a set S, a topological space X and an injective mapping from S to X. This gave us a topology on S: the induced topology. In this chapter we shall consider a topological space X, a set Y and a sul)ective mapping from X to Y. This will give us a topology on Y: the so-called 'quotient' topology
- Topological spacesA topological space is just a set together with certain subsets (which will be called open sets) satisfying three properties
- Product spacesOur final general method of constructing new topological spaces from old ones is through the direct product. Recall that the direct product X X Y of two sets X,Y is the set of ordered pairs (x,y) with x from X and y from Y. If X and Y are topological spaces we can use the topologies on X and Y to give one on X x Y
- Compact spacesWe shall look at properties of spaces that are preserved under homeomorphisms. One important consequence of this is that if one space has the property in question and another does not, then the two spaces cannot be homeomorphic
- Hausdorff spacesAll metrizable spaces are Hausdorff, in particular with the usual topology and any space with the discrete topology is Hausdorff. A space with the concrete topology is not Hausdorff if it has at least two points
- Connected spacesIntuitively a space X is connected if it is in 'one piece'; but how should a 'piece' be interpreted topologically? It is reasonable to require that open or closed subsets of a 'piece' are open or closed respectively in the whole space X. We will see that a 'piece' is an open and closed subset of X
