On pseudocompactness of remainders of certain spaces

Abstract

Let ${\mathcal B}$ be
a base for a nowhere locally compact
Tychonoff space $X$ and let $bX$ be a compactification of $X$.
Then the following two statements hold:

\begin{enumerate}
\item The remainder $bX\setminus X$ of $X$ is pseudocompact
if and only if for any countable infinite subfamily ${\mathcal V}$ of
${\mathcal B}$ there exists an accumulation point of the family
${\mathcal V}$ in $bX\setminus X$.

\item If for any countable
infinite subfamily ${\mathcal V}$ of
${\mathcal B}$ the set of all accumulation points of the family ${\mathcal V}$ in
$X$ is not a nonempty compact set of $X$, the
$bX\setminus X$ is pesudocompact.
\end{enumerate}

Let $X=\prod_{i\in I}X_{i}$ be a product
space and $S$ be a subset of $X$ satisfying the following
condition:

$(\ast)$ For each nonempty countable set $J\subset I$, the projection $p_{J}:X\rightarrow\prod_{i\in J}X_{i}$
satisfies that $p_{J}(S)=X_{J}:=\prod_{i\in J}X_{i}$.
\\If $\mathcal{B}$ is the canonical base for $X$ and
$\mathcal{V}_S=\{B_{i}\cap S:i\in\omega\}$ is a countable
infinite subfamily of $\mathcal{B}_S=\{B\cap S:B\in\mathcal{B}\}$
such that the set $F$ of all accumulation points of the family
$\mathcal{V}_{S}$ in $S$ is nonempty, then for any $a\in F$ there
exists a countable subset $J$ of $I$ such that
$p_{J}^{-1}(p_{J}(a))\cap S=p_{J}^{-1}(p_{J}(a))\cap F$ and for
any $\alpha\in I\setminus J$, $p_{\alpha}(F)=X_{\alpha}$.

By the above conclusions, we can get two known results in
\cite{TATK}. We finally show that if $X=\prod_{i\in I}X_{i}$ is a
product of a family $\{X_{i}:i\in I\}$ of Tychonoff spaces such
that uncountably many of them are non-compact and $Y$ is a dense
subspace of $X$, then for every compactification $bY$ of $Y$ the
remainder $bY\setminus Y$ is pseudocompact.