Abstract
The technique of extension of compact spaces by means of adding suprema of continuous functions was introduced by Koszmider in the construction of an indecomposable Banach space of the form C(K). Generally, the behaviour of such extensions with respect to connectedness is delicate. We provide conditions under which any extension is connected, as well as showing that connectedness cannot be preserved indefinitely, i.e., that for every metrizable compactum K there exists a disconnected space L which is obtained by finitely many extensions starting with K.
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Notes
A compact Hausdorff space is quasi-Stonean if the closure of every open \(F_\sigma \) is open.
References
Engelking, R.: General topology, \(2^a\)ed. Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin (1989)
Fajardo, R.: An indecomposable Banach space of continuous functions which has small density. Fund. Math. 202(1), 43–63 (2009)
Fajardo, R.: Quotients of indecomposable Banach spaces of continuous functions. Stud. Math. 212(3), 259–283 (2012)
Koszmider, P.: Banach spaces of continuous functions with few operators. Math. Ann. 300, 151–183 (2004)
Koszmider, P.: A space \(C(K)\) where all nontrivial complemented subspaces have big densities. Stud. Math. 168(2), 109–127 (2005)
Koszmider, P.: On large indecomposable Banach spaces. J. Funct. Anal. 364(8), 1779–1805 (2013)
Koppelberg, S.: General theory of boolean algebras. In: Monk, J.D. (ed.) Handbook of Boolean Algebras. Elsevier Science Publishers B.V, Amsterdam (1989)
Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer, Berlin (1991)
Plebanek, G.: A construction of a Banach space \(C(K)\) with few operators. Topol. Appl. 143, 217–239 (2004)
Stone, M.: Boundedness properties in function-lattices. Can. J. Math. 1, 176–186 (1949)
Veksler, A.I., Geiler, V.A.: Order completeness and disjoint completeness of linear partially ordered spaces. Sib. Math. 13, 43–51 (1972). (Russian)
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The authors wish to acknowledge the helpful comments of the referee.
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Supported by a Grant from CAPES, Number 1328310, and a Grant from CNPQ, Number 142058/2016-5.
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Barbeiro, A.S.V., Fajardo, R.A.d.S. Suprema of continuous functions on connected spaces. São Paulo J. Math. Sci. 11, 189–199 (2017). https://doi.org/10.1007/s40863-016-0055-3
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DOI: https://doi.org/10.1007/s40863-016-0055-3