On Mutual Compactificability and Compactificability Classes

conference paper

English

\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 $\Bbb R$.

compact space, compactification, $\theta$-regular space

KOVÁR, M.

23. 8. 1996

Topology Atlas

173

177

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http://at.yorku.ca/p/p/a/b/04.htm