Topological spaces are generalizations of metric space topologies

up:: Metric Topology

The definition of Topology is inspired by the properties of all metric-induced topologies. Instead of thinking about open sets as being generated by Open Balls (Metric Spaces), one thinks about them as being elements of a subset of the power set .

Motivating properties for topological spaces

1. Intersection of finitely many open sets is also open

Let . Then .

Let . Then

Let . Then

2. Union of arbitrarily many open sets is also open

Let an arbitrarily-large set of open sets. Then .

Let . Then

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References

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