Math 2.0

General topology Definition

Basis of a topology

Builds on Topological spaceUnion of a family of sets

Definition

Let (X,τ)(X, \tau) be a topological space. A basis of τ\tau is a set ℬ⊆τ\mathcal{B} \subseteq \tau such that each U∈τU \in \tau is a union of members of ℬ\mathcal{B}. Equivalently, ℬ⊆τ\mathcal{B} \subseteq \tau and for each U∈τU \in \tau and each x∈Ux \in U there is B∈ℬB \in \mathcal{B} with x∈B⊆Ux \in B \subseteq U.

Remarks

A family ℬ\mathcal{B} of subsets of a set XX is a basis of some topology on XX if and only if two conditions hold. First, the members of ℬ\mathcal{B} cover XX. Second, for all B1,B2∈ℬB_1, B_2 \in \mathcal{B} and x∈B1∩B2x \in B_1 \cap B_2 there is B3∈ℬB_3 \in \mathcal{B} with x∈B3⊆B1∩B2x \in B_3 \subseteq B_1 \cap B_2. This topology is then unique. The open intervals form a basis of the usual topology of ℝ\mathbb{R}. The intervals with rational endpoints form a countable basis. In a metric space, the open balls form a basis of the metric 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 levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionTopological space, definition, no evidence levelTopological spacedefinitionBasis of a topology, definition, no evidence levelBasis of a topologydefinition
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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. Union of a family of sets definition
  16. Topological space definition
  17. Basis of a topology definition

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35 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Base_(topology), Definition and basic properties
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Tag06EX0FHBTF
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Kinddefinition
WritingW1 stated
MSC54A05
DefinesBasis of a topology
UsesTopological space, Union of a family of sets

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