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 $\mathbb{P}(X)$.

Motivating properties for topological spaces

1. Intersection of finitely many open sets is also open

Let $A,B\in {\tau }_d$. Then $A\cap B\in {\tau }_d$.

Let $x\in A\cap B$. Then

• $\exists r_{x,A}\mid x\in B_{r_{x,A}}(x)\subset A$
• $\exists r_{x,B}\mid x\in B_{r_{x,B}}(x)\subset B$

Let $r\equiv \min (r_{x,A},r_{x,B})$. Then $x\in B_r(x)\subset A\cap B\square$

2. Union of arbitrarily many open sets is also open

Let $\{A_i{\}}_{i\in I}\subset {\tau }_d$ an arbitrarily-large set of open sets. Then $\bigcup _{i\in I}{A}_i\in {\tau }_d$.

Let $x\in \bigcup _{i\in I}{A}_i$. Then $\exists k\in I\mid x\in A_k⟹\exists r_k\mid x\in B_{r_k}(x)\subset A_k\subset \bigcup _{i\in I}{A}_i\square$

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References

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