Math 2.0

General topology Definition

Limit point

Builds on Topological spaceOpen set

Definition

Let XX be a topological space, let S⊆XS \subseteq X and let x∈Xx \in X. The point xx is a limit point of SS when each open set UU with x∈Ux \in U contains a point of SS other than xx.

Remarks

The point xx need not be in SS. In ℝ\mathbb{R}, the point 00 is a limit point of {1/n:n≥1}\{1/n : n \ge 1\}. In a metric space, xx is a limit point of SS if and only if each ball B(x,ε)B(x, \varepsilon) contains a point of S∖{x}S \setminus \{x\}. The closure of SS is the union of SS and the set of its limit points. A point of SS that is not a limit point of SS is an isolated point of SS.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, 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 setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionLimit point, definition, no evidence levelLimit pointdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Topological space definition
  22. Function definition
  23. Metric space definition
  24. Open ball definition
  25. Metric topology definition
  26. Open set definition
  27. Limit point definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Accumulation_point, Definition
About this page
Tag0HBQC2GJXT
URIm2://0HBQC2GJXT
Kinddefinition
WritingW1 stated
MSC54A05
DefinesLimit point
UsesTopological space, Open set

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