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Math and science::Topology

Compactness. Examples

  • The collection of sets {(n1,n+1):nZ} form an open cover of R. This cover has [no finite subcover/a finite subcover?].
  • Divide the interval [0,1] like so: U0=[0,12], Un=(2n,1] for n1, then form the cover (Un)n0. This is an open cover, and has many finite subcovers (e.g. ([0, 1/2), (1/2, 1], (1/4, 1]). So we can say that [0,1] is [...].
  • The compact subspaces of Rn are the [...].
  • Any indiscrete space is [compact/not compact, possibly with conditions].
  • Any finite space is [compact/not compact, possibly with conditions].
  • A discrete space is [compact/not compact, possibly with conditions].
  • In a normed vector space V, the closed unit ball is [compact/not compact, possibly with conditions].