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\begin{center}
{\LARGE On the comparison between\\
closure spaces and topological spaces}
\vspace{10mm}
\begin{flushright}
{\large 教科・領域教育専攻\\
自然系コース(数学)\\
西\ 村\ 雅\ 彦\mbox{\ \ \ \ }}
\end{flushright}
\end{center}
\vspace{0mm}]


%%%%%%%%%%%%%%%%%%%%  概要　本文　　%%%%%%%%%%%%%%%%%%%%%%%%%%
Through the text books due to J.~Nagata [1] and E.~\v{C}ech [3] and related papers cited
 in the last page, I have studied and surveyed the classes of both topological spaces and
 closure spaces. Especially, in this survey the comparison between completely regular
 spaces and closure spaces from the viewpoint of embedding, uniformities and proximities
 is cleared. The results are what are known already by the experts rather than new. Here
 topological spaces and closure spaces are defined in terms of closure operator, as follows:\\
For T-space;\\
\noindent {\bf 1.19 Proposition}[see [1], Proposition II.8.B]
Let $R$ be a set and $A$ be a subset of $R$. We define a function $A\rightarrow \overline{A}$ from
 the family of a set $R$ to it satisfying the following conditions:\\
%Let $A$ and $B$ be a subset of a topological space $R$. We denoted the closure
 of $A$ by $%\overline{A}$. Closure satisfies the following conditions:\\
i)\hspace{2mm} $\overline{\emptyset}=\emptyset $,\\
ii)\hspace{2mm} $\overline{A}\supset A$,\\
iii)\hspace{2mm} $\overline{A\cup B}=\overline{A}\cup \overline{B}$,\\
iv)\hspace{2mm} $\overline{\overline{A}}=\overline{A}$.\\
We shall say that the function is a $closure$ $operator$ and $\overline{A}$ is
 called $closure$ $of$ $A$. A space defined this operator is called 
 a $topological$ $space$ or $T$-$space$.\\
\noindent For closure space;\\
\noindent {\bf 4.1\ Definition}[[3],Definition 14.A.1]
If $P$ is a set and $u$ is a single-valued relation on $exp P$ ranging in $exp P$, then
 we shall say that $u$ is a $closure$ $operation$ (or simply a $closure$) for $P$ provided
 that the following conditions (also called axioms) are satisfied:\\
(cl 1)\hspace{2mm} $u\emptyset =\emptyset $,\\
(cl 2)\hspace{2mm} $X\subset uX$ for each $X\subset P$,\\
(cl 3)\hspace{2mm} $u(X\cup Y)=uX\cup uY$ for each $X\subset P$ and $Y\subset P$.

The main results are summarized in Theorems 1.33, 2.7, 3.16, 4.46 and 5.15 mainly.

A topological space and closure space are characterized for embedding the follows:\\
For T-space;\\
\noindent {\bf 1.33\ Theorem}[[1],Theorem III.1]
A topological space $R$ is completely regular space if and only if it is homeomorphic
 with a subspace of the product space of copies of the unit segment $[0,1]$.\\

\noindent For closure space;\\
\noindent {\bf 4.46\ Theorem}[[3],Theorem 17.C.18]
Suppose that $P=(y_1,y_2,y_3)$ is a three-point set and $u$ is a closure for $P$ such that 
\begin{eqnarray*}
u(y_1)=(y_1,y_2),\ \ u(y_2)=u(y_3)=P.
\end{eqnarray*}
(Obviously there exists exactly one such closure $u$.) Then every closure 
space $\langle Q,v\rangle $ is homeomorphic with a subspace of the product
 space $\langle P,u\rangle ^{expQ}$.\\

With respect to uniformity, there is a similarity between T$_{3\frac 12}$-spaces and
 closure spaces.\\
For T-space;\\
\noindent {\bf 2.7\ Theorem}[[1],Theorem VI.13]
A topological space $R$ has a uniformity compatible with its topology if and
 only if $R$ is completely regular.\\

\noindent For closure space;\\
\noindent {\bf 5.15\ Theorem}[[3],Theorem 23.B.3]
In order that a closure operation $u$ for a set $P$ be semi-uniformizable it is
 necessary and sufficient that $x\in u(y)$ imply $y\in u(x)$, i.e. if $x$ belongs to
 the closure of a one-point set $(y)$, then $y$ belongs to the closure of $(x)$.\\

With respect to proximity,\\
For T-space;\\
\noindent {\bf 3.16\ Theorem}[[1],Theorem VI.20]
Let $R$ be a topological space. Then we can introduce a proximity compatible
 with the topology of $R$ if and only if $R$ is completely regular.\\

\noindent For closure space, by Theorem 6.21 and Proposition 6.18 
every \v{C}ech proximity $\delta $ is induced by a semi-uniformity $\calu $, and
 that among all semi-uniformities inducing a given proximity $\delta $ there exists
 a unique \v{C}ech proximally coarse semi-uniformity $\calu $. Hence a closure space
 has a \v{C}ech proximity compatible with it if and only if it is semi-uniformizable
 and if and only if for each point $x,y$ in it, $x\in c(y)\rightarrow y\in c(x)$.\\

The undefined terms such as uniformity, proximity, \v{C}ech proximity are referred
 to the corresponding definitions in the article.\\

\noindent {\bf Reference}\\

\noindent [1] J. Nagata, {\it Modern General Topology}, 1974, North-Holland Publishin Co, Amsterdam. Chap.I-VI.\\

\noindent [2] R.Engelking, {\it General Topology}, 1989, Heldermann Verlag, Berlin.\\

\noindent [3] E. \v{C}ech,  {\it Topological Spaces}, 1966, John Wiley \& Sons. Chap.III,IV\\

\noindent [4] A.Mysior, {\it A regular space which is not completely regular}, American Mathematical Society 81 (1981) 652-653.\\

\vspace{4cm}
\begin{center}
{\large 指導\ \ 　溝上　武實}
\end{center}
\end{document}