## ABSTRACT

This chapter introduces the reader to the various properties of Euclidean spaces. These properties are used in later chapters and also help in the understanding of proofs of theorems that deal with the capabilities of biomimetic neural networks.

The chapter begins with a brief introduction of the set of real numbers R, starting with Dedekind cuts and concluding with pertinent topological properties of R. The remainder of the chapter addresses two matters: the first pertains to topological properties of n-dimensional Euclidean spaces R
^{n}
, while the second is concerned with the role of Euclidean spaces in artificial intelligence with emphasis on pattern recognition and artificial neural networks. The major highlights in the discussion of n-dimensional Euclidean space are the notions of bounded sets, continuity, compactness, connectedness, the Bolzano-Weierstrass theorem, and the Heine-Borel theorem. The final section of the chapter explores the notion of artificial intelligence (AI), explains the mathematical definition of patterns and pattern recognition, and provides some examples of pattern recognition. The section concludes with a brief discussion of artificial neural networks (ANN) and the rationale of biomimetic neural networks (BNN).