Iterated reduced cluster functions

Abstract

Given a multifunction $F$ between topological spaces $X$ and $Y$, the reduced cluster function $C^r(F;\cdot): X \rightarrow 2^Y$ of $F$ is defined by $C^r(F;x)=\bigcap$ cl$(F(U\setminus \{x\}))$ running through the neighborhood system of $x$. By transfinite recursion, one defines iterated reduced cluster functions $C^{r,\alpha}(F;\cdot)$ for all ordinals $\alpha > 0$.

We characterize multifunctions $F$ that are invariant in the sense of $C^r(F;\cdot)=F$. For every countable ordinal $\alpha$, we describe the family of all iterated reduced cluster functions $C^{r,\alpha}(F;\cdot)$ of arbitrary multifunctions $F: X \rightarrow 2^Y$ and the family of all iterated reduced cluster functions $C^{r,\alpha}(f;\cdot)$ of arbitrary functions $f: X \rightarrow Y$, provided that $X$ and $Y$ are metrizable spaces and $Y$ is separable.