# Hewitt realcompactification

*Hewitt compactification, Hewitt extension*

An extension of a topological space that is maximal relative to the property of extending real-valued continuous functions; it was proposed by E. Hewitt in [1].

A homeomorphic imbedding $\nu : X \rightarrow Y$ is called a functional extension if $\nu(X)$ is dense in $Y$ and if for every continuous function $f : X \rightarrow \mathbb{R}$ there exists a continuous function $\bar f : Y \rightarrow \mathbb{R}$ such that $f = \bar f \nu$. A completely-regular space $X$ is called a *Q*-space or a functionally-complete space if every functional extension of it is a homeomorphism, that is, if $\nu(X) = Y$. A functional extension $\nu : X \rightarrow Y$ of a completely-regular space is called a Hewitt extension if $Y$ is a *Q*-space. A completely-regular space has a Hewitt extension, and the latter is unique up to a homeomorphism.

The Hewitt extension can also be defined as the subspace of those points $y$ of the Stone–Čech compactification $\beta X$ for which every continuous real-valued function $f : X \rightarrow \mathbb{R}$ can be extended to $X \cup \{y\}$.

#### References

[1] | E. Hewitt, "Rings of real-valued continuous functions, I" Trans. Amer. Math. Soc. , 64 (1948) pp. 45–99 |

[2] | R. Engelking, "Outline of general topology" , North-Holland (1968) (Translated from Polish) |

[3] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) |

#### Comments

The Hewitt extension is not a compactification, hence the phrase "Hewitt compactification" is rarely used.

**How to Cite This Entry:**

Hewitt realcompactification.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Hewitt_realcompactification&oldid=34336