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Published in:
2011 | OriginalPaper | Chapter
1. Topological Spaces
Author : Gregory L. Naber
Published in: Topology, Geometry and Gauge fields
Publisher: Springer New York
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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 \((\|x\| = {({({x}^{1})}^{2} + \cdots+ {({x}^{n})}^{2})}^{1/2}).\)