Detail publikace

On mutual compactificability of topological spaces

KOVÁR, M.

Originální název

On mutual compactificability of topological spaces

Anglický název

On mutual compactificability of topological spaces

Jazyk

en

Originální abstrakt

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

Anglický abstrakt

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

Dokumenty

BibTex


@inproceedings{BUT3375,
  author="Martin {Kovár}",
  title="On mutual compactificability of topological spaces",
  annote="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",
  address="Matematicko-fyzikální fakulta Univerzity Karlovy",
  booktitle="Abstracts of the Eight Prague Topological Symposium",
  chapter="3375",
  institution="Matematicko-fyzikální fakulta Univerzity Karlovy",
  year="1996",
  month="august",
  pages="55",
  publisher="Matematicko-fyzikální fakulta Univerzity Karlovy",
  type="conference paper"
}