Math and science::Topology

Metric space. ε-balls

Let $X$ be a metric space, let $x \in X$ and let $ε > 0$ be a real. The open ε-ball around $x$ (or in more detail, the open ball around $x$ of radius $ε$ ) is the subset of $X$ given by

B (x, ε) = {y \in X : d (x, y) < ε}

Similarly, the closed ε-ball around $x$ is

\bar{B} (x, ε) = {y \in X : d (x, y) \leq ε}

The definition of open sets in metric spaces is formulated in terms of open ε-balls.

Example

Metrics such as $d_{1}$ , $d_{2}$ and $d_{\infty}$ for both $R^{n}$ and continuous function spaces $C [a, b]$ , and other metrics such as Hamming distance, each induce ε-balls within the space they are applied in. It is interesting to consider what the ε-ball in each metric space is.

Metric space. ε-balls

Example

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