The metric topology is generated by open balls

up:: Metric Topology

Let $(M,d)$ be a Metric Space, and $U\subset M$ an open set.

Then, for each $x\in U$, there is some open ball $B_{r_x}(x)$ such that $x\in B_{r_x}(x)\subset U$.

Thus, $\bigcup _{x\in U}{B}_{r_x}(x)\subset U$. Consequently, it is equal to $U$, since it contains all of its points and not any more.

Thus, any open set $U\subset M$ is a union of open balls. I.e. the set of all open balls in $M$ is a Topological Basis for $M$‘s Metric Topology.

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References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 24
• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 27