Topology

up:: Topological Space

A topology over a set $X$ is a set $\tau$ such that

1. $\varnothing ,X\in \tau$
2. Algebraically closed under arbitrary unions: $\forall \{B_i{\}}_{i\in \Lambda }\subset \tau ,\bigcup _{i\in \Lambda }{B}_i\in \tau$
3. Algebraically closed under finite intersections: $\forall \{B_i{\}}_{i=1}^n\subset \tau ,\bigcap _{i=1}^n{B}_i\in \tau$

These properties were motivated by the properties satisfied by Metric Topology’s.

Examples of topologies

For a given set $X\ne \varnothing$, some elementary topologies are:

• Trivial (chaotic) topology ${\tau }_t=\{\varnothing ,X\}$
• Discrete topology (powerset) ${\tau }_d=\mathbb{P}(X)$

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References

• Sutherland, Wilson A. Introduction to metric and topological spaces. Oxford University Press, 2009.
• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 27