## Abstract

Our goal in this chapter is to show that most of the concepts introduced in the previous chapters for the set ℝ of real numbers can be extended to any abstract *metric space*; i.e., a set on which the concept of *metric* (or *distance*) can be defined. Indeed, as we have already seen, the basic concept of *limit* which we studied in Chapters 2 and 3, and used to define (in Chapter 4) the related concept of continuity, is defined in terms of *distance*. Let us recall that the distance between two real numbers *x* and *y* is defined to be *d*(*x*, *y*) := |*x* - *y*|, and satisfies three simple properties: For any numbers *x*, *y*, *z* ∈ ℝ we have: (i) |*x* - *y*| ≥ 0, and equality holds if and only if *x* = *y*; (ii) |*x* - *y*| = |*y* − x|; and (iii) |x − *y*| ≤ |*x* − *z*| + |*y* − *z*|. Property (iii) is called the *Triangle Inequality* for obvious geometrical reasons. Using this distance, we defined, in Chapter 2, the concepts of ε-neighborhood, open set, closed set, limit point, isolated point, convergent sequence and Cauchy sequence. We then defined the concept of limit for general real-valued functions of a real variable, and proved that such limits can also be defined in terms of limits of sequences. Also, before introducing the related notion of *continuity*, we introduced (in Chapter 4) the *topological* concepts of *compactness* and *connectedness*. All these notions can be defined, in essentially the same way, in any (abstract) *metric space* and the proofs of most theorems are basically copies of the ones we gave for the special metric space ℝ if one replaces |*x* − *y*| by *d*(*x*, *y*) throughout. Many proofs will therefore be brief or will be left as exercises for the reader.

## Keywords

Limit Point Open Ball Open Cover Cauchy Sequence Connected Subset## Preview

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