Clopen Subsets of X* and 2 - Homeomorphic Spaces



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In this thesis, we shall give characterizations of, and other results about, clopen subsets of X* under various conditions on X. We shall use as our main tools pseudo-clopen sets (a generalization of a clopen set), pseudo-disconnections (a generalization of a disconnection using pseudo-clopen, rather than clopen, sets), and trivial autohomeomorphisms (auto- homeomorphisms of X* that arise as extensions of certain homeomorphisms of X). Among other results, we are able to characterize the clopen subsets of X* when X is nowhere locally compact, characterize when X* is disconnected in the cases when X is locally compact, nowhere locally compact, or metrizable, and develop conditions on X that guarantee when an autohomeomorphism of X* will extend to points of X. We also deal extensively with 2 - homeomorphic spaces, a recent topic initially studied by Arhangel'skii and Maksyuta. We study 2 - homeomorphism as a relation (giving the result that every infinite space is 2 - homeomorphic to some space to which it is not homeomorphic), and we are able to characterize the spaces that are 2 - homeomorphic to a compact space, and to a discrete space. We show that being scattered and having finite Cantor - Bendixson height are both preserved under 2 - homeomorphism, and characterize the spaces that are 2 - homeomorphic to n copies of the one-point compactification of a discrete space.