Throughout this paper the spaces $X_i$ are assumed
Tycho\-noff, $(X_I)_\kappa$ denotes $X_I:=\Pi_{i\in I}\,X_i$
with the $\kappa$-box topology (so $(X_I)_\omega=X_I$),
and $\Sigma_\lambda(p):=\{x\in X_I:|\{i\in I:x_i\neq p_i\}|<\lambda\}$
whenever $p\in X_I$.

Milton Ulmer proved in 1970/1973 that
in a product of
first-countable spaces, each $\Sigma$-product is $C$-embedded. The authors
generalize this theorem in several ways.

{\bf Theorem}. Let $\kappa$ be a regular cardinal.
If $q\in(X_I)_\kappa\backslash\Sigma_{\kappa^+}(p)$ and each $q_i$
is a $P(\kappa$)-point in $X_i$ with $\chi(q_i,X_i)\leq\kappa$, then
$\Sigma_{\kappa^+}(p)$ is $C$-embedded in $\Sigma_{\kappa^+}(p) \cup \{q\}$.

{\bf Corollary}. Let $\kappa$ be a regular cardinal. If
each $X_i$ is a $P(\kappa)$-space such that $\chi(X_i)\leq\kappa$,
then in $(X_I)_\kappa$ each set of the form $\Sigma_{\kappa^+}(p)$
is $C$-embedded.

In the same works, Ulmer constructed an example showing that
a $\Sigma$-product in a product of spaces of countable pseudocharacter
need not be $C$-embedded. Again the authors modify and extend
his work, this time as follows.

{\bf Theorem}. For every $\kappa\geq\omega$
there are a set $\{X_i:i \in I\}$
of Tychonoff spaces with $|I|=\kappa$,
$q \in X_I$ and $f \in C(X_I\backslash\{q\},\{0,1\})$
such that no continuous function from $X_I$
to $[0,1]$ extends $f$.
One may arrange further that $|X_i|=\kappa$ for every $i\in I$, and either

\,(i) there is $i_0\in I$ such that
$\psi(X_{i_0}) = \omega$ and for $i_0\neq i\in I$ the space
$X_i$ is the one-point compactification of a discrete space with
cardinality $\kappa$; or

(ii) the spaces $X_i$ are pairwise homeomorphic, with $\psi(X_i)=\omega$,
and:

~~~~~either (a) all but one point in each $X_i$ is isolated, or
(b) each $X_i$ is dense-in-itself.