Topological Spaces

Abstract

We begin by recording a few items from real analysis (our canonical reference for this material is [Sp1],Chap. 1 –Chap. 3, which should be consulted for details as the need arises). For any positive integer n, Euclidean n-space ${\mathbb{R}}^{n} =\{ ({x}^{1},\ldots,{x}^{n}) :\ {x}^{i} \in\mathbb{R},\ i = 1,\ldots,n\}$ is the set of all ordered n-tuples of real numbers with its usual vector space structure (x + y = (x ¹, …, x ⁿ) + (y ¹, …, y ⁿ) = (x ¹ + y ¹, …, x ⁿ + y ⁿ) and ax = a(x ¹, …, x ⁿ) = (ax ¹, …, ax ⁿ)) and norm $(\|x\| = {({({x}^{1})}^{2} + \cdots+ {({x}^{n})}^{2})}^{1/2}).$