Math 2.0

General topology Definition

Topological space

Builds on SubsetUnion of a family of setsIntersectionEmpty set

Definition

Let XX be a set. A topology on XX is a set τ\tau of subsets of XX with these three properties:

  1. The empty set ∅\varnothing and the set XX are in τ\tau.
  2. For each family (Ui)i∈I(U_i)_{i \in I} of members of τ\tau, the union ⋃i∈IUi\bigcup_{i \in I} U_i is in τ\tau.
  3. For all U,V∈τU, V \in \tau, the intersection U∩VU \cap V is in τ\tau.

A topological space is a pair (X,τ)(X, \tau) where XX is a set and τ\tau is a topology on XX.

Remarks

The members of τ\tau are the open sets of the space. Their complements are the closed sets. By induction, property 3 gives that each finite intersection of open sets is open. Infinite intersections need not be open. Each metric gives a topology, the metric topology. Other basic examples are the discrete topology and the indiscrete topology. One can also define a topology by its closed sets, or by the neighbourhoods of its points.

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To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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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 spacedefinition
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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

Where this leads

291 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Topological_space, Definition via open sets
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