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Math and science::Topology

Metric space. ε-balls

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

[ \( B(x, \varepsilon) = \{y \in X : \text{what condition?} \} \) ]

Similarly, the closed ε-ball around \( x \) is

[\( \bar{B}(x, \varepsilon) = \{y \in X : \text{what condition?} \} \) ]