Topology of ℝn
As indicated at the conclusion of the last chapter, we cannot expect ℝn to behave as a particularly interesting algebraic system for n ≠ 1, 2, 3, 4, 8. On the other hand, if we look instead at the ‘continuity’ properties of ℝn (to be precise, its topology), we have a very rich situation, which we attempt in this chapter to sketch. More than a sketch is quite impossible, since topology is a vast branch of mathematics, increasing rapidly both in itself and in its applications to mathematics and physics ; and the interested reader should turn to the books described in Section 25.14 for detailed information. However, the basic ideas of topology are essential for an educated understanding of elementary mathematics, and they are quite accessible to the reader of this book, as we now show. We begin with some motivating remarks.
KeywordsTopological Space Triangle Inequality Quotient Space Jordan Curve Projective Geometry
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