Continuous functions on extremally disconnected spaces

Author:
J. Vermeer

Journal:
Trans. Amer. Math. Soc. **347** (1995), 3263-3285

MSC:
Primary 54G05; Secondary 54C15, 54H25

DOI:
https://doi.org/10.1090/S0002-9947-1995-1311920-0

MathSciNet review:
1311920

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using results and techniques due to Abramovich, Arenson and Kitover it is shown that each fixed-point set of a selfmap of a compact extremally disconnected space is a retract of that space, and that the retraction can be constructed from the particular selfmap itself. Also, the closure of the set of periodic points turns out to be a retract of the space. Several decomposition theorems for arbitrary selfmaps on extremally disconnected spaces are obtained similar to the theorem of Frolík on embeddings. Conditions are obtained under which the set of fixed points is clopen.

**[A,A,K]**Y.A. Abramovich, E.L. Arenson and A.K. Kitover,*Banach**-modules and operator preserving disjointness*, Pitman Res. Notes in Math. Ser., no. 277, Essex, England.**[B,F]**B. Balcar and F. Franek,*Universal minimal dynamical systems for discrete semigroups*, manuscript.**[B,K]**Aleksander Błaszczyk and Dok Yong Kim,*A topological version of a combinatorial theorem of Katětov*, Comment. Math. Univ. Carolin.**29**(1988), no. 4, 657–663. MR**982783****[D]**Eric K. van Douwen,*𝛽𝑋 and fixed-point free maps*, Topology Appl.**51**(1993), no. 2, 191–195. MR**1229715**, https://doi.org/10.1016/0166-8641(93)90152-4**[D,M]**Eric K. van Douwen and Jan van Mill,*Subspaces of basically disconnected spaces or quotients of countably complete Boolean algebras*, Trans. Amer. Math. Soc.**259**(1980), no. 1, 121–127. MR**561827**, https://doi.org/10.1090/S0002-9947-1980-0561827-0**[F]**Z. Frolík,*Fixed points of maps of extremally disconnected spaces and complete Boolean algebras*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**16**(1968), 269–275 (English, with Loose Russian summary). MR**0233343****[H,V]**K.P. Hart and J. Vermeer,*Homeomorphisms on**-spaces*(to appear). [K] A.K. Kitover,*Operators preserving disjointness and the mappings of extremally disconnected compact spaces*, Optimizatsiya**40**(1987), 138-143.**[K]**Kim Do Young, Ph.D. thesis.**[K,S]**Adam Krawczyk and J. Steprāns,*Continuous colourings of closed graphs*, Topology Appl.**51**(1993), no. 1, 13–26. MR**1229497**, https://doi.org/10.1016/0166-8641(93)90011-2**[vM]**Jan van Mill,*An introduction to 𝛽𝜔*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 503–567. MR**776630****[V]**J. Vermeer,*Continuous maps on finite dimensional spaces*(manuscript).**[V]**J. Vermeer,*Frolík’s theorem for basically disconnected spaces*, Acta Univ. Carolin. Math. Phys.**34**(1993), no. 2, 135–142. Selected papers from the 21st Winter School on Abstract Analysis (Poděbrady, 1993). MR**1282976****[Wal]**Russell C. Walker,*The Stone-Čech compactification*, Springer-Verlag, New York-Berlin, 1974. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83. MR**0380698**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
54G05,
54C15,
54H25

Retrieve articles in all journals with MSC: 54G05, 54C15, 54H25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1311920-0

Keywords:
Extremally disconnected,
retracts,
fixed points

Article copyright:
© Copyright 1995
American Mathematical Society