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What is a compactification?

What is a compactification? Here are some definitions.

Noun
  1. (topology) Any of various procedures of enlarging a topological space to make it compact.
  2. (topology) The space resulting from any such procedure.
  3. (physics) Any modification of a theory such that an infinite parameter becomes finite
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Examples
They relate Boolean algebras to general topology and to the theory of rings and ideals, and include what is called Stone-tech compactification today.
The most studied superstring compactification is heterotic string theory compactified on a Calabi-Yau space in six-dimensions.

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