A note to classes of mutual comapctificability II

This contribution partly completes my talk presented on Prague Topological
Symposium in 1996.

\mezerka

\comment
A 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.
%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.
\endcomment

\definition{Definition 1} 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 extending the topologies of $X$
and $Y$ to the union $K=X\cup Y$ such that any two points $x\in X$, $y\in Y$ have
disjoint neighborhoods in $K$. If, in addition, the topology on $K$
can be Hausdorff, we say that $X$ is {\it $T_2$-compactificable} by $Y$ or
that $X$, $Y$ are {\it mutually $T_2$-compactificable}.
\enddefinition

\definition{Definition 2} Let $\Top$ be the class of all topological spaces.
For any $X,Z \in \Top$ we define $X\thicksim Z$ if for every non-empty space $Y\in \Top$
the space $X$ is compactificable by $Y$ if and only if $Z$ is compactificable by $Y$.
It can be easily seen that $\thicksim$ is an equivalence relation on $\Top$. Let
us denote by $\C(X)$ the equivalence class in $\Top$ with respect to $\thicksim$
containing $X$ and call it the {\it compactificability class} of $X$.
Now, for any $X,Z\in \Top$ we put $\C(X)\gre\C(Z)$ if for every non-empty space $Y\in \Top$
it holds if the space $X$ is compactificable by $Y$ then $Z$ is compactificable by $Y$.
Obviously, the relation $\gre$ is an order on the class $\C(\Top)=\left\{\C(X)|X\in \Top
\right\}$. If for some $X,Z\in \Top$ it holds $\C(X)\gre\C(Z)$ but $\C(X)\ne\C(Z)$ we
write $\C(X)\gr\C(Z)$. The classes of {\it $T_2$-compactificability} can be defined
analogously.

\mezerka

The relation $\gre$ between the compactificability classes can be
interpreted as some scale for various kinds of ``non-compactness''.
In addition to general theorems, the compactificability classes are also
tested on some familiarly known spaces, e.g. the spaces derived from the
Cantor space or the real line. In comparison with my talk in Prague in 1996,
some new results will be presented.

\mezerka

{\bf What~do~you~think?} \ \
Let $K=\left<0,1\right>^3$ be the unit closed cube. Let
$X=\left(0,1\right)\times\left(0,1\right)\times\left\{0\right\}$ and
$Y=K\smallsetminus Y$. Of course, X, Y are mutually compactificable and
$X$ is homeomorphic to $\R^2$. Can one replace $X$, for example, by the real
ray $\left<0, \infty\right)$ or by the real line $\R$?