论文标题
2022年夏季分析任意关闭
Arbitrarily Close for Summer 2022 Analysis
论文作者
论文摘要
无论如何，分析的内核是以下想法：如果点的每个社区与集合相交，则一个点是任意接近集合的。以这种方式定义``任意关闭''为微积分和真实分析的经典结果提供了基础，该结果涉及融合，限制，连接性，限制，连续性，差异，分化，集成，系列等。本书包含：任意关闭的详尽介绍；使用任意关闭作为关键第一步的序列限制和收敛的方法；欧几里得空间的拓扑结构是由封闭式组成的；探索域，范围和集合和序列的函数的性质；以及欧几里得空间之间连续功能的基本方面的介绍。即便如此，``任意关闭''比这里讨论的要深。这个想法本质上是拓扑结构，还有更多探索。
The kernel of analysis, to me anyway, is the following idea: A point is arbitrarily close to a set if every neighborhood of the point intersects the set. Defining ``arbitrarily close'' in this way provides a foundation for classical results in calculus and real analysis dealing with convergence, limits, connectedness, limits, continuity, differentiation, integration, series, and more. This book contains: a thorough introduction to arbitrarily close; an approach to limits and convergence of sequences using arbitrarily close as a key first step; the topology of Euclidean spaces stemming from closed sets; an exploration of the properties of functions like domains, ranges, and images of sets and sequences; and an introduction to basic aspects of continuous functions between Euclidean spaces. Even so, ``arbitrarily close'' reaches deeper than discussed here. The idea is topological in nature and there is much more to explore.
