Topological Spaces

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) < r}. In $\mathbb{R}$, $B(x,r) = (x - r,x + r)$. A set O is called open in a metric space if for each x ∈ O there is an r > 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, $\left \Vert z\right \Vert =\Vert z - 0\Vert = d(z,0)$.