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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.

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 inuitive notion of open sets should find these examples familiar.

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:

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: