The video defines compactness by stating that a set E is compact if, for every nested sequence of closed nonempty subsets E ⊃ E₁ ⊃ E₂ ⊃ E₃ ⊃ ⋯, there exists at least one element that belongs to all of the Eₙ. This condition ensures that the sets do not escape to infinity.
Conditions: E is a set.; Each Eₙ is a closed subset of E.; Each Eₙ is nonempty.; The sequence is nested by inclusion: E ⊃ E₁ ⊃ E₂ ⊃ ⋯.
The video defines compactness by stating that a set E is compact if, for every nested sequence of closed nonempty subsets E ⊃ E₁ ⊃ E₂ ⊃ E₃ ⊃ ⋯, there exists at least one element that belongs to all of the Eₙ. This condition ensures that the sets do not escape to infinity.
Conditions: E is a set.; Each Eₙ is a closed subset of E.; Each Eₙ is nonempty.; The sequence is nested by inclusion: E ⊃ E₁ ⊃ E₂ ⊃ ⋯.