Open Balls (Metric Spaces)

up:: Metric Space

One can think about the definition of an open ball in a metric space as the set of all points equidistant (with respect to the space’s Metric Function) to some base point.

Given a point $x\in X\ne \varnothing$, one can think about the open ball of radius $r$ around $x$ as

$$B_r(x)=\{y\in X\mid d(x,y)<r\}$$

With this definition of open balls, one can think about Open Sets in Metric Spaces, which are sets which, for each point it contains, also contain some open ball around each of them. In this sense, open balls are the generators of open sets.

Examples of $d_1,d_{p_1\in (1,2)},d_2,d_{p_2>2},d_{\infty }$ in ${\mathbb{R}}^2$:

Generalization to Topological Spaces

Open balls offer the motivation for the idea of Topology: instead of thinking about open balls as the generators of all open sets, one starts from the very set of all open sets itself, which is a subset of the power set $\mathbb{P}(X)$.

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References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 24