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所在平台: Udemy |
课程主页: https://www.udemy.com/course/real-analysis-part-1/
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**课程名称：** 实分析（一）：基础拓扑 **课程概述：** 本课程是实分析系列的第一部分，专注于基础拓扑概念。实分析是研究实数、实数序列、级数以及实值函数行为的数学分支。本课程将深入探讨实值序列和函数的关键性质，包括收敛性、极限、连续性、光滑性、微分和积分。 **核心内容：** * **集合论基础：** 介绍集合的基本概念。 * **映射：** 深入理解一对一映射（Injective）、满射映射（Surjective）、双射映射（Bijective）的概念。 * **可数集与不可数集：** 详细介绍可数集、不可数集和最多可数集的定义，并提供相应的例子。 * **可数集与不可数集定理：** 讲解与可数集和不可数集相关的定义及重要定理。 * **开集与闭集：** 定义开集和闭集，并探讨它们的交集和并集性质，以及基于此的定理。 * **邻域：** 详细解释邻域的定义，并提供相关定理和详细的说明。 * **开集、闭集与开闭集（Clopen）定理：** 探讨开集、闭集以及开闭集（既是开集也是闭集的集合）的定义、定理和性质。 * **极限点、内点、外点：** 定义极限点、内点、外点，并提供相关的定理和示例。 * **稠密集：** 介绍稠密集的定义和相关定理。 * **完美集：** 定义完美集及其相关概念。 * **完美集定理：** 讲解完美集相关的定理。 * **关键定理：** 深入阐述波尔查诺-魏尔斯特拉斯定理（Bolzano Weierstrass theorem）和海涅-博雷尔定理（Heine Borel theorems），包含所有预期定理、作业和相关示例。 **欢迎加入本课程，一起探索实分析和基础拓扑的精彩内容！**
Course Introduction:In the mathematics world, Real analysis is the branch of mathematics analysis that studies the behavior of real numbers, sequences and series of real number and real-valued function. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiation, and integrability.Key content of the course:Basic Topology introduction to setsConcept of One one mapping, Onto mapping, Injective and surjective mappingsCountable sets, uncountable sets and atmost countable sets with examples.Definition and Theorems on countable sets and uncountable setsDefinition of Open sets and closed setsIntersection and union of sets and theorem based on intersection and union of closed and open setDefinition of Neighborhood of a pointDetailed explanation and theorems on Neighborhood of pointTheorems on Open sets and closed sets and Clopen setsDefinition of Limit points, Interior points, exterior pointsTheorems and examples on limit points, interior points and exterior pointsDense sets and Theorems on Dense setsDefinition and concept of Perfect setsTheorems on perfect setsBolzano Weierstrass theoremHeine Borel theorems including all expected Theorems, Assignments and Examples of above contents.So enroll the course and explore the content. See you on the course!
