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''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 [[#References|[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|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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hewitt,   "Rings of real-valued continuous functions, I"  ''Trans. Amer. Math. Soc.'' , '''64'''  (1948)  pp. 45–99</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  R. Engelking,   "Outline of general topology" , North-Holland  (1968)  (Translated from Polish)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR>
</table>



====Comments====
The Hewitt extension is not a [[Compactification|compactification]], hence the phrase  "Hewitt compactification"  is rarely used.

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