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Algebraic characterizations of some disconnected spaces. (English) Zbl 1117.54030

The authors characterize various disconnectedness properties ((strong) zero-dimensionality, basical or extremal disconnectedness) by means of ideals in \(C(X)\), mostly that they are generated by some idempotents. Another characterization: \(X\) is extremally disconnected iff \(C(X)\) is a Baer ring iff \(C^*(X)\) is a Baer ring (for P-spaces one gets regular Baer rings).

MSC:

54C40 Algebraic properties of function spaces in general topology
16S60 Associative rings of functions, subdirect products, sheaves of rings
54D99 Fairly general properties of topological spaces
54F65 Topological characterizations of particular spaces
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