Topological Spaces

Peter A. Loeb
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Peter A. Loeb: University of Illinois

Chapter Chapter 9 in Real Analysis, 2016, pp 147-178 from Springer

Abstract: Abstract In this chapter, we generalize the notion of closeness given by a metric or norm. Recall that if we start with a metric space (X, d), then open balls play a central role in obtaining results. An open ball with center x and radius r > 0 is denoted by B(x, r); it is the set {y ∈ X: d(x, y) 0 such that the open ball B(x, r) is contained in O. If z ∈ B(x, r), then there is an open ball $$B(z,\delta ) \subseteq B(x,r)$$ . Just let $$\delta = r - d(x,z)$$ . This property makes an open ball such as B(x, r) an open set. A subset of a metric space is again a metric space when the metric is restricted to pairs of points in the subset. In a space X with a norm, such as a Hilbert space, the corresponding metric is given by $$d(x,y) = \left \Vert x - y\right \Vert$$ . Note that for any z ∈ X, $$B(0,r) + z = B(z,r)$$ . Also, for any z ∈ X, $$\left \Vert z\right \Vert =\Vert z - 0\Vert = d(z,0)$$ .

Keywords: Local Filter Base; Uniform Closure; Pointwise Closure; Disjoint Open Neighborhoods; Trivial Topology (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/978-3-319-30744-2_9

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