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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 ⊃ ⋯.

The real number line is not compact because one can construct a nested sequence of closed nonempty subsets whose intersection is empty. For example, taking Eₙ = [n, $\infty$) creates intervals that shift rightward indefinitely, leaving no common point.

Conditions: The ambient space is the real number line ℝ.; Compactness is defined via nested closed nonempty subsets having a common point.; Eₙ = [n, $\infty$) for $n = 1$, 2, 3, ...

The real number line is not compact because one can construct a nested sequence of closed nonempty subsets whose intersection is empty. For example, taking Eₙ = [n, $\infty$) creates intervals that shift rightward indefinitely, leaving no common point.

Conditions: The ambient space is the real number line ℝ.; Compactness is defined via nested closed nonempty subsets having a common point.; Eₙ = [n, $\infty$) for $n = 1$, 2, 3, ...