Hranice množiny

Množina (světle modře) a její hranice (tmavě modře)

Hranice množiny je v matematice pojem z topologie značící množinu všech takových prvků, jehož každé okolí obsahuje alespoň jeden bod zadané množiny a alespoň jeden bod mimo zadanou množinu. Značí se ∂M. Podobný význam má hranice metrického prostoru a hranice variety s hranicí.

Formální definice v topologii

Existují tři ekvivalentní definice hranice množiny (X je univerzum):

  • Hranice množiny ∂M je uzávěr množiny M bez vnitřku množiny M: ∂M = M \ M°.
  • Hranice množiny ∂M je průnik uzávěru M s uzávěrem jejího doplňku: ∂M = M ∩ (X \ M).
  • Hranice množiny ∂M je množina všech bodů b univerza X takových, že každé okolí b obsahuje alespoň jeden bod patřící do M a alespoň jeden bod patřící do X \ M.

Příklady

Mějme množinu reálných čísel R s běžnou topologií založenou na otevřených intervalech. Pak:

Poslední dva příklady ukazují, že hranice husté množiny s prázdným vnitřkem je jejím uzávěrem[pozn 1].

  • V množině racionálních čísel s topologií otevřených intervalů, hranice množiny , kde a je iracionální, je prázdná množina[pozn 2].

Reference

  • John L. Kelley: General topology, Birkhäuser, 1975
  • James Munkres: Topology, Cambridge University Press, 2nd edition, 1988

Poznámky

  1. Budeme postupovat podle první definice. Pro hustou množinu platí, že její uzávěr je celé univerzum. A neboť vnitřkem je prázdná množina, tak hranicí je celé univerzum.
  2. Stačí si uvědomit, že ani jeden z hraničních bodů tohoto intervalu neleží v množině racionálních čísel.

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