Real analysis

Real analysis is a field in mathematics that focuses on the set of real numbers, their properties, sequences and functions. Included in this branch of mathematics is concepts of limits and convergence, calculus, and properties of real-valued functions such as continuity. It also includes measure theory.

For the purposes of this article, "analysis" will be limited to the generalization and extension of the concepts of calculus, using the concepts of elementary point-set topology.

The reader should be quite familiar with the concepts of calculus, especially limits and continuity. In particular, the reader should be comfortable with the ramifications of the phrase "for every epsilon".

Open sets

Open sets (and, by extension, closed sets, which are just the complements of open sets) are the fundamental concept of analysis. Analysis and topology are really just the study of open sets.

Before giving the definition of open sets in Euclidean space, we present some examples. Readers who are aware of the general intuitive notion of open sets should find these examples familiar.

The simplest open sets in 1-dimensional Euclidean space (formally called <math>\mathbb{R}^1</math>; informally called the real numbers of the "real line") are open intervals. An open interval consists of those numbers lying strictly between two endpoints a and b. In set-theoretic notation:

<math>\{ x\ |\ a < x < b \}\,</math>

A shorter notation for this set consists of the two endpoints in parentheses:

<math>( a, b )\,</math>

A closed interval (we will have more to say about closed sets later) would include the endpoints. It is commonly denoted with brackets:

<math>[ a, b ] = \{ x\ |\ a \le x \le b \}</math>

An interval that includes one endpoint but not the other is called semi-open:

<math>[ a, b ) = \{ x\ |\ a \le x < b \}</math>

<math>( c, d ] = \{ x\ |\ c < x \le d \}</math>

When drawing pictures of intervals, those same symbols are typically used:

Need a picture here!

Open intervals are not the only open sets. Any union of open intervals is an open set. For example:

<math>\{ N, N+1/N\ |\ N \textrm{\ is\ an\ integer} \ge 2 \}</math>

Bizarrely defined sets like the one above are commonly used as examples and counterexamples in analysis and topology.