## Real Analysis Exchange

### Iterated reduced cluster functions

Christian Richter

#### 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.

#### Article information

Source
Real Anal. Exchange, Volume 30, Number 1 (2004), 43-58.

Dates
First available in Project Euclid: 27 July 2005

https://projecteuclid.org/euclid.rae/1122482115

Mathematical Reviews number (MathSciNet)
MR2126793

Zentralblatt MATH identifier
1068.54023

#### Citation

Richter, Christian. Iterated reduced cluster functions. Real Anal. Exchange 30 (2004), no. 1, 43--58. https://projecteuclid.org/euclid.rae/1122482115

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