 ##  [Stone-Čech Compactification](/stone-cech-compactification) 

  ##  [Stone-Čech Compactification](https://mathlogic.quantumdictionary.io/stone-cech-compactification-0) 

  
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- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

The (up to unique homeomorphism) largest compactification βX of a completely regular (Tychonoff) space X characterized by the property that every bounded continuous real-valued function on X extends uniquely to a continuous function on βX.

 
 ##  [Stone-Čech Compactification](https://puremath.quantumdictionary.io/stone-cech-compactification-1) 

  
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- Mathematics &amp; Logic -

**Pure Mathematics Dictionary**

 
Definition

For a completely regular (Tychonoff) space X, the Stone‑Čech compactification βX is a compact Hausdorff space equipped with a dense embedding i: X → βX characterized by the universal property that every continuous map from X to any compact Hausdorff space K extends uniquely to a continuous map βX → K.