On Mutual Compactificability and Compactificability Classes

\abstract
Conceived intuitively, various topological spaces undoubtedly have different
degrees of ``compactness'' or ``non-compactness''. But how to
practically determine whether some space is more non-compact than the other?

In this work the main criterion of the ``level of non-compactness'' of a topological space
$X$ is its ability to form, together with another space $Y$,
a compact space $K=X\cup Y$ such that the points of $X$ are in $K$ separated
from the points of $Y$ by disjoint open neighbourhoods.

Noticing that the existence of such topology on $K$ implies $\theta$-regularity of
both $X$ and $Y$, at the background of these considerations lies the idea to
imagine the compact space as a box of bricks or jigsaw puzzle where ``bricks''
or ``pieces'' are certain $\theta$-regular spaces.
The principal problem is which ``pieces'' are so compatible that they together
can create some compact space. For simplicity, accepting the jigsaw
model, in this work we will deal with puzzles of two pieces. Since finding a
general applicable criterion of such compatibility of $\theta$-regular spaces
(``bricks'' or ``pieces'') is a difficult problem, we will
proceed indirectly. We will compare different spaces in their behaviour with
respect to the above criterion and join together to one class the spaces whose
behaviour is similar.

After testing on the real line, we obtain that any non-compact locally connected metrizable
generalized continuum as well as Cantor space without its ``left corner'' $\Bbb
D^{\aleph_0}\smallsetminus \left\{\boldkey 0\right\}$ are of the same class
of mutual compactificability as $\Bbb R$.