부산대학교 도서관

In a topological semigroup $S$ the closure of a subset $A$ is denoted by $[A]$, the product of two subsets $A$ and $B$ by $AB$ and for every $s\in S$ the component and the quasicomponent of $s$ are denoted by $C_s$ and $Q_s$, respectively. If $A$ is a closed subsemigroup of $S$ and $\alpha$ an ordinal number the powers ${}^\alpha\!A$, $\overset{\alpha}\to{A}$ and $A^\alpha$ are defined 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 limit ordinal $\alpha$. 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 modulo the component $C_0$. This fact implies the following: If $S$ is also regular then $Q_0=C_0$. If $S$ is such that for every $s\in S$ there exists an integer $n>1$ for which $s^n=s$ then again $Q_0=C_0$. In the 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 semigroup of all continuous selfmaps $f\colon X\rightarrow X$ and $S_2(X)$ its subsemigroup 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 respect to which $K$ becomes a semitopological semigroup, the quasicomponents in $K$ are one-point sets. In particular, this applies in the case where $X$ has the discrete topology and $S(X)$ the topology of pointwise convergence.