부산대학교 도서관

In a topological semigroup $S$ the closureof a subset $A$ is denoted by $[A]$, the productof two subsets $A$ and $B$ by $AB$ and for every $s\in S$the component and the quasicomponent of $s$ are denotedby $C_s$ and $Q_s$, respectively. If $A$ is a closedsubsemigroup of $S$ and $\alpha$ an ordinal number the powers${}^\alpha\!A$, $\overset{\alpha}\to{A}$ and $A^\alpha$ aredefined inductively as follows: $A^1=\overset{1}\to{A}={}^1\!A=A$; $A^{\alpha+1}=[(A^\alpha)A]$; $\overset{\alpha+1}\to{A}=[(\overset{\alpha}\to{A})(\overset{\alpha}\to{A})]$;${}^{\alpha+1}A=[A(^\alpha A)]$ and $A^\alpha=\bigcap_{\beta<\alpha} A^\beta$;$\overset{\alpha}\to{A}=\bigcap_{\beta<\alpha}\overset{\beta}\to{A}$;${}^\alpha A=\bigcap_{\beta<\alpha}{}^\beta A$ for a limitordinal $\alpha$. Given two closed subsemigroups $A$ and$B$ of $S$, the subsemigroup $A$ is said to be transfinitely$r$-nilpotent [transfinitely $l$-nilpotent, transfinitely nilpotent]modulo $B$ if there exists a transfinite ordinal $\alpha$ for which$A^\alpha\subseteq B$ $[^\alpha A\subseteq B$, $\overset{\alpha}\to{A}\subseteq B]$. The authors show that if $S$ has a zero element $0$then the quasicomponent $Q_0$ is transfinitely $r$-nilpotent modulothe component $C_0$. This fact implies the following: If $S$ is alsoregular then $Q_0=C_0$. If $S$ is such that for every $s\in S$ thereexists an integer $n>1$ for which $s^n=s$ then again $Q_0=C_0$. Inthe case that $S$ is homogeneous and regular then $Q_s=C_s$ for every$s\in S$. For a topological space $X$ let $S(X)$ denote the semigroupof all continuous selfmaps $f\colon X\rightarrow X$ and $S_2(X)$ itssubsemigroup consisting of those $f$ for which the cardinality of$f(X)$ is $\le2$. It is proved that if $K$ is any subsemigroup of $S(X)$containing $S_2(X)$ then for any $T_1$-topology on $K$ with respectto which $K$ becomes a semitopological semigroup, the quasicomponentsin $K$ are one-point sets. In particular, this applies in the casewhere $X$ has the discrete topology and $S(X)$ the topology of pointwiseconvergence.