Math 2.0

General topology Definition

Subbasis

Builds on Topological spaceBasis of a topologyIntersection of a family of sets

Definition

Let XX be a set and let ๐’ฎ\mathcal{S} be a set of subsets of XX. The topology generated by ๐’ฎ\mathcal{S} is the smallest topology on XX that contains ๐’ฎ\mathcal{S}. It is the intersection of all topologies on XX that contain ๐’ฎ\mathcal{S}. If (X,ฯ„)(X, \tau) is a topological space and ฯ„\tau is the topology generated by ๐’ฎ\mathcal{S}, then ๐’ฎ\mathcal{S} is a subbasis of ฯ„\tau.

Remarks

The open sets of the generated topology are the unions of finite intersections of members of ๐’ฎ\mathcal{S}. Here the empty intersection is XX. Thus the finite intersections of members of ๐’ฎ\mathcal{S} form a basis of the generated topology. The open rays (โˆ’โˆž,a)(-\infty, a) and (b,โˆž)(b, \infty), with a,bโˆˆโ„a, b \in \mathbb{R}, form a subbasis of the usual topology of โ„\mathbb{R}. The sets ฯ€iโˆ’1(U)\pi_i^{-1}(U), with UU open, form a subbasis of the product topology.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionIntersection of a family of sets, definition, no evidence levelIntersection of a famโ€ฆdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of โ€ฆdefinitionTopological space, definition, no evidence levelTopological spacedefinitionBasis of a topology, definition, no evidence levelBasis of a topologydefinitionSubbasis, definition, no evidence levelSubbasisdefinition
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  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Predicate definition
  8. Empty set object
  9. Family of sets definition
  10. Intersection definition
  11. Union definition
  12. Existential quantifier definition
  13. Universal quantifier definition
  14. Subset definition
  15. Intersection of a family of sets definition
  16. Union of a family of sets definition
  17. Topological space definition
  18. Basis of a topology definition
  19. Subbasis definition

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8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Subbase, Definition
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Tag0CHR35YNJQ
URIm2://0CHR35YNJQ
Kinddefinition
WritingW1 stated
MSC54A05
DefinesSubbasis
UsesTopological space, Basis of a topology, Intersection of a family of sets

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