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.