Poznámka k vzájemné kompaktifikovatelnosti II

A note to classes of mutual comapctificability II

conference paper

en

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 $\R$?

Různé nekompaktní topologické prostory mají různý stupeň "nekompaktnosti". V této práci zkoumáme schopnost topologického prostoru X vytvořit s jiným, disjunktním topologickým prostorem Y kompaktní prostor, v němž dva různé body z nichž jeden leží v X a druhý v Y mají disjunktní okolí.

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 $\R$?

topological space as a jigsaw (compactness versus non-compactness, $\theta$-regularity, compactification, Cantor spaces, subspaces of $\R$

09.08.1998

János Bolyai Mathematical Society

Abstarcts of the Topology Conference in Gyula

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