COMPACTIFICATION Meaning and
Definition
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Compactification is a mathematical concept utilized in various branches of mathematics, including topology, algebraic geometry, and differential geometry. It refers to a process of adding extra points or structures to a given space or object to make it more "compact" or complete.
In general terms, a compactification takes an existing space, often non-compact, and results in a modified space that contains additional points or structures, making it behave like a compact space. These additional points or structures are introduced in a systematic manner so that certain desired properties are preserved.
One typical example of compactification is the process of adding points at infinity to the real number line to form the extended real line. This compactification ensures that every sequence of real numbers has a limit, even if it tends to infinity.
Compactifications often play a significant role in the study of geometric and topological properties of spaces. They allow mathematicians to analyze spaces with infinite or unbounded elements by transforming them into compact spaces, where techniques from compact space theory can be applied. Compactifications also provide a framework to study limit behaviors, continuity, and convergence, among other important concepts.
Overall, compactification is a powerful tool in mathematics that enriches and enhances the study of spaces, enabling deeper investigations and applications that may not be readily accessible in the original space.
Common Misspellings for COMPACTIFICATION
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Etymology of COMPACTIFICATION
The word "compactification" is derived from the term "compact", which has its roots in Latin. The Latin word "compactus" means "compressed" or "close together". In mathematics, a compact set refers to a set that is closed and bounded, capturing the idea of being "compressed" or "close together" in a general sense.
Adding the suffix "-tion" to "compact" forms the noun "compactification". This suffix is commonly used to indicate the act or process of making something compact, such as the formation or creation of a compact object or structure.
Therefore, the term "compactification" in mathematics refers to the process of modifying or extending a given space to make it compact, typically by "closing off" or adding additional points or elements to the existing space.
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