On mutual compactificability of topological spaces

Recall topological space $X$ is said to be {\it $\theta$-regular} \cite{Ja}
if every filter base in $X$ with a $\theta$-cluster point has a cluster point.
In Hausdorff spaces, $\theta$-regularity coincides with regularity. Further
properties of $\theta$-regular spaces are also studied in \cite{Ko}.
Through this work, $\theta$-regularity plays a fundamental role.
A topological space is said to be ({\it strongly}) {\it locally compact}
if every $x\in X$ has a compact (closed) neighborhood. Compactness
is regarded without any separation axiom.
The following concepts will be introduced:

\definition{Definition} Let $X$, $Y$ be topological spaces with $X\cap
Y=\varnothing$. The space $X$ is said to be {\it compactificable} by the
space $Y$ or, in other words, $X$, $Y$ are called {\it mutually compactificable}
if there exists a compact topology on $K=X\cup Y$ extending the topologies of $X$
and $Y$ such that any two points $x\in X$, $y\in Y$ have
disjoint neighborhoods in $K$. If, in addition, there exists a Hausdorff
topology on $K$ extending the topologies of $X$, $Y$ we say that $X$ is {\it
$T_2$-compactificable} by $Y$ or that $X$, $Y$ are {\it mutually
$T_2$-compactificable}.
\enddefinition

\example{\it Preliminary observations}
(i) A real interval is $T_2$-compactificable by any real interval.
(ii) A discrete space is $T_2$-compactificable by a copy of itself.
(iii) A space is compactificable by a finite discrete space iff the
space is strongly locally compact.
(iv) For a space $X$ there exist a space $Y$ which is $T_2$-compactificable by
$X$ iff $X$ is $T_{3{1\over 2}}$.
\endexample

We intend to discuss some variants the of concepts defined above and also some
of the following natural questions:

\roster
\item"(1)" Characterize all topological spaces $X$ such that there exists a space $Y$
such that $X$, $Y$ are mutually compactificable.
\item"(2)" Characterize those topological spaces $X$ that are ($T_2$-) compactificable by
some fixed space $Y$.
\item"(3)" Characterize those topological spaces that are ($T_2$-) compactificable by a copy
of itself.
\endroster