Notes - Metric Spaces MT23, Balls

Created: July 20, 2023

Define the open ball $B(a, \varepsilon)$.

\[B(a, \varepsilon) = \\{x \in X : d(x, a) < \varepsilon\\}\]

Define the closed ball $\bar B(a, \varepsilon)$.

\[B(a, \varepsilon) = \\{x \in X : d(x, a) \le \varepsilon\\}\]

Suppose $X$ is a metric space and $Y \subseteq X$. How can you write $B _ Y(a, \varepsilon)$?

\[B_Y(a, \varepsilon) = Y \cap B_X(a, \varepsilon)\]

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