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The Heine-Borel Theorem states that in finite-dimensional Euclidean space ℝⁿ, a subset is compact if and only if it is both closed and bounded. This provides a practical criterion for identifying compact sets in standard geometric spaces.

Conditions: The space is finite-dimensional Euclidean space ℝⁿ.; S is a subset of ℝⁿ.

The Heine-Borel Theorem states that in finite-dimensional Euclidean space ℝⁿ, a subset is compact if and only if it is both closed and bounded. This provides a practical criterion for identifying compact sets in standard geometric spaces.

Conditions: The space is finite-dimensional Euclidean space ℝⁿ.; S is a subset of ℝⁿ.