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Readers who are aware of the general intuitive notion of open sets should find these examples familiar. An open interval consists of those numbers lying strictly between two endpoints a and b. In set-theoretic notation:.
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A closed interval we will have more to say about closed sets later would include the endpoints. It is commonly denoted with brackets:. Open intervals are not the only open sets. Any union of open intervals is an open set. For example:.
Bizarrely defined sets like the one above are commonly used as examples and counterexamples in analysis and topology. In two or more dimensions the situation becomes more complicated, because even simple open sets can come in an endless variety of shapes. The fundamental open set equivalent to an open interval is the open neighborhood , also called an open ball.
An open neighborhood has a center point and a nonzero radius, and is the set of all points whose distance from the center is strictly less than that radius. The double-stroke absolute value sign is the norm or the metric distance function. The double-stroke absolute value sign is similar to the usual absolute value operation, generalized to arbitrary dimensions or other metric spaces.
It is easy to see that, in two dimensions, an open neighborhood is the interior of a circle. This point is crucial—the whole subject of analysis and topology depends on it! To draw a picture of an open neighborhood, use a circle bounded by a dotted line. To make a closed ball, the formula would be:. But open neighborhoods are the important sets from a theoretical standpoint.
This construction is shown in the diagram at the right. Here are a few extremely fundamental and far-reaching theorems. Some of them are surprisingly simple:. Theorem: Any union of open sets, including unions of an infinite number of open sets, is an open set. That neighborhood must be contained in the union.
This theorem can be extended for any finite intersection, but it does not work for infinite intersections. Here is an example:. Proof: We need a neighborhood of each point in the space. The neighborhood centered on that point, with radius 1, will do the trick. This means that the real line is open. The real line is an open set , because it is:.
Infinite unions are allowed, even though infinity is not a number. Proof: It contains a neighborhood of each of its points; those are its constituent neighborhoods. Throughout this book, we will begin to see that we do not need intuition to understand mathematics - we need a manual.
The overarching thesis of this book is how to define the real numbers axiomatically. How would that work? This book will read in this manner: we set down the properties which we think define the real numbers. We then prove from these properties - and these properties only - that the real numbers behave in the way which we have always imagined them to behave. We will then rework all our elementary theorems and facts we collected over our mathematical lives so that it all comes together, almost as if it always has been true before we analyzed it; that it was in fact rigorous all along - except that now we will know how it came to be.
Lecture 10A - Real Analysis 1
Do not believe that once you have completed this book, mathematics is over. In other fields of academic study, there are glimpses of a strange realm of mathematics increasingly brought to the forefront of standard thought. After understanding this book, mathematics will now seem as though it is incomplete and lacking in concepts that maybe you have wondered before. In this book, we will provide glimpses of something more to mathematics than the real numbers and real analysis. After all, the mathematics we talk about here always seems to only involve one variable in a sea of numbers and operations and comparisons.
Note: A table of the math symbols used below and their definitions is available in the Appendix. A select list of chapters curated from other books are listed below. They should help develop your mathematical rigor that is a necessary mode of thought you will need in this book as well as in higher mathematics. This part of the book formalizes the various types of numbers we use in mathematics, up to the real numbers. This part focuses on the axiomatic properties what we have defined to be true for the sake of analysis of not just the numbers themselves but the arithmetic operations and the inequality comparators as well.
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This part of the book formalizes the definition and usage of graphs, functions, as well as trigonometry. The most curious aspect of this section is its usage of graphics as a method of proof for certain properties, such as trigonometry. These methods of proof are mostly frowned upon due to the inaccuracy and lack of rigorous definition when it comes to graphical proofs , but they are essential to derive the trigonometric relationships, as the analytical definition of the trigonometric functions will make using trigonometry too difficult—especially if they are described early on.
The following chapters will rigorously define the trigonometric functions. They should only be read after you have a good understanding of derivatives, integrals, and inverse functions. This part of the book formalizes sequences of numbers bound by arithmetic, set, or logical relationships. This part focuses on concepts such as mathematical induction and the properties associated with sets that are enumerable with natural numbers as well as a limit set of integers.
This part of the book formalizes the concept of distance in mathematics, and provides an introduction to the analysis of metric space. This part of the book formalizes the concept of intervals in mathematics, and provides an introduction to topology.
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