Real Analysis Exchange

Iterated reduced cluster functions

Christian Richter

Full-text: Open access


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

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

First available in Project Euclid: 27 July 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54C50: Special sets defined by functions [See also 26A21] 54C60: Set-valued maps [See also 26E25, 28B20, 47H04, 58C06]
Secondary: 54C08: Weak and generalized continuity

cluster set reduced cluster set reduced cluster function of order α


Richter, Christian. Iterated reduced cluster functions. Real Anal. Exchange 30 (2004), no. 1, 43--58.

Export citation


  • G. Aumann, Reelle Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 68, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1954.
  • E. F. Collingwood, Cluster set theorems for arbitrary functions with applications to function theory, Ann. Acad. Sci. Fenn. Ser. A I, 336/8 (1963), 15 pp.
  • R. Engelking, General topology, Sigma Series in Pure Mathematics 6, Heldermann Verlag, Berlin, 1989.
  • H. Hahn, Theorie der reellen Funktionen, Band I, Verlag Julius Springer, Berlin, 1921.
  • H. Hahn, Reelle Funktionen, Erster Teil: Punktfunktionen, Akademische Verlagsgesellschaft m.b.H., Leipzig, 1932, reprinted by Chelsea Publishing Company, New York, 1948.
  • J. E. Joseph, Regularity, normality and multifunctions, Proc. Amer. Math. Soc., 70 (1978), 203–206.
  • J. E. Joseph, Multifunctions and cluster sets, Proc. Amer. Math. Soc., 74 (1979), 329–337.
  • A. S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics 156, Springer-Verlag, Berlin, Heidelberg, New York, 1994.
  • C. Richter, The cluster function of single-valued functions, submitted.
  • B. S. Thomson, Real functions,
  • J. D. Weston, Some theorems on cluster sets, J. London Math. Soc., 33 (1958), 435–441.