Math 2.0

General topology Proposition

Epsilon-net characterisation of total boundedness

Builds on Totally bounded setOpen ball

Statement

Let (X,d)(X, d) be a metric space and let S⊆XS \subseteq X. Then SS is totally bounded if and only if for each real ε>0\varepsilon > 0 there is a finite set F⊆SF \subseteq S with S⊆⋃y∈FB(y,ε).S \subseteq \bigcup_{y \in F} B(y, \varepsilon).

Remarks

Such a finite set FF is an ε\varepsilon-net of SS. Here B(y,ε)B(y, \varepsilon) is an open ball. Proof: an ε\varepsilon-net gives a cover by balls of diameter at most 2ε2\varepsilon. Conversely, from a cover by sets AkA_k of diameter less than ε\varepsilon, choose one point of SS in each AkA_k that meets SS. The centres can be taken in SS or in XX. Both choices give the same notion.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionExtended real number line, object, no evidence levelExtended real number …objectProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSupremum, definition, no evidence levelSupremumdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionDiameter of a set, definition, no evidence levelDiameter of a setdefinitionTotally bounded set, definition, no evidence levelTotally bounded setdefinitionEpsilon-net characterisation of total boundedness, statement, no evidence levelEpsilon-net character…statement
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. Extended real number line object
  3. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. Set definition
  7. Implication definition
  8. Predicate definition
  9. Supremum definition
  10. Ordered pair definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Metric space definition
  19. Open ball definition
  20. Diameter of a set definition
  21. Totally bounded set definition
  22. Epsilon-net characterisation of total boundedness statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Totally_bounded_space, In metric spaces
About this page
Tag0NMGJJ19HD
URIm2://0NMGJJ19HD
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 0NMGJJ19HD.
UsesTotally bounded set, Open ball

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.