Continuous functions on extremally disconnected spaces

Vermeer, J.

doi:10.1090/S0002-9947-1995-1311920-0

Continuous functions on extremally disconnected spaces
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Trans. Amer. Math. Soc. 347 (1995), 3263-3285 Request permission

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.

References

Banach

-modules and operator preserving disjointness

Universal minimal dynamical systems for discrete semigroups

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
Eric K. van Douwen, $\beta X$ and fixed-point free maps, Topology Appl. 51 (1993), no. 2, 191–195. MR 1229715, DOI 10.1016/0166-8641(93)90152-4
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, DOI 10.1090/S0002-9947-1980-0561827-0
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 Russian summary). MR 233343

Homeomorphisms on

-spaces

Operators preserving disjointness and the mappings of extremally disconnected compact spaces

40

Adam Krawczyk and J. Steprāns, Continuous colourings of closed graphs, Topology Appl. 51 (1993), no. 1, 13–26. MR 1229497, DOI 10.1016/0166-8641(93)90011-2
Jan van Mill, An introduction to $\beta \omega$, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 503–567. MR 776630

Continuous maps on finite dimensional spaces

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
Russell C. Walker, The Stone-Čech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83, Springer-Verlag, New York-Berlin, 1974. MR 0380698