Math 2.0

Manifolds and cell complexes Surfaces Example

Möbius strip

Builds on Quotient topology

Example

Let ∼\sim be the equivalence relation on the square [0,1]×[0,1][0, 1] \times [0, 1] generated by (0,y)∼(1,1−y)(0, y) \sim (1, 1 - y) for all y∈[0,1]y \in [0, 1]. The quotient space [0,1]2/∼[0, 1]^2 / {\sim}, with the quotient topology, is the Möbius strip.

Remarks

The gluing has a half-twist: the bottom of the left edge is glued to the top of the right edge. The Möbius strip is a non-orientable surface. Its boundary is one circle, not two. Without the twist, the gluing (0,y)∼(1,y)(0, y) \sim (1, y) gives a cylinder, which is not homeomorphic to the Möbius strip.

Reading path

To understand this page from the ground up, read these 31 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 setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionSet-builder notation, concept, no evidence levelSet-builder notationconceptExistential 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 spacedefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionSymmetric relation, definition, no evidence levelSymmetric relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionEquivalence relation, definition, no evidence levelEquivalence relationdefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionEquivalence class, definition, no evidence levelEquivalence classdefinitionQuotient set, definition, no evidence levelQuotient setdefinitionQuotient topology, definition, no evidence levelQuotient topologydefinitionMöbius strip, example, no evidence levelMöbius stripexample
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. Set definition
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Predicate definition
  8. Empty set object
  9. Family of sets definition
  10. Ordered pair definition
  11. Intersection definition
  12. Union definition
  13. Set-builder notation concept
  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. Reflexive relation definition
  23. Symmetric relation definition
  24. Transitive relation definition
  25. Function definition
  26. Equivalence relation definition
  27. Preimage of a set definition
  28. Equivalence class definition
  29. Quotient set definition
  30. Quotient topology definition
  31. Möbius strip example

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/M%C3%B6bius_strip, lead section
  2. Web https://en.wikipedia.org/wiki/M%C3%B6bius_strip, Properties
About this page
Tag04AJE0M271
URIm2://04AJE0M271
Aliasesmobius-band
Kindexample
WritingW2 usable
MSC57K20
Instance ofTopological space
UsesQuotient topology
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