Ivan Shishkin, Rye (1878)

Problems/Compact setReviewed

Escape from infinity

by AilDesOurs·translated by Ancient Tree·
44
Difficulty scaleÉchelle de difficulté

This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
·
English
EnglishFrançais

Let EE be a real normed vector space of infinite dimension, and let K⊂EK \subset E be compact.

  1. Show that E\KE \backslash K is path-connected.

  2. Give a counterexample in finite dimension.

Note: this result shows that compact subsets of an infinite-dimensional space cannot separate the space into different regions. There is always a way around them!

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References

  1. Serge Francinou, Hervé Gianella, Serge Nicolas — Oraux X-ENS — Analyse 3
Details

D'après Oraux X-ENS, analyse 3, S.Francinou,H.Gianella,S.Nicolas, Cassini

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