Open Sets in Metric Spaces

up:: Open Balls (Metric Spaces)

Given a Metric Space $(X,\tau )$ and some set $A\subset X$, then $A$ is said to be open if, for all points $x\in A$, there exist some open ball $B_{r_x}(x)$ with radius $r_x$ dependent on $x$.

The set of all open sets in a metric space (as defined above) is called the Metric Topology, or metric-induced topology.

Generalization to Topological Spaces

Instead of thinking about open sets as being dependent on the metric structure of a space (i.e. containing some open ball around its points), a Topology is already the set which contains all open sets from the get-go.

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References

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