On mutual compactificability of topological spaces

conference paper

en

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

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

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

19.08.1996

Matematicko-fyzikální fakulta Univerzity Karlovy

Abstracts of the Eight Prague Topological Symposium

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