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A compact Hausdorff topology that is a \(T_1\)-complement of itself. (English) Zbl 1020.54002
Two topologies \(\tau_1\) and \(\tau_2\) on a set \(X\) are called \(T_1\)-complementary if their intersection is the cofinite topology and their union is a subbase for the discrete topology on \(X\) . Topological spaces \((X,\tau)\) and \((Y,\sigma)\) are called \(T_1\)-complementary provided that there exists a bijection \(f: X \rightarrow Y\) such that \(\tau\) and \(\{f^{-1}(U) : U \in \sigma \}\) are \(T_1\)-complementary topologies on \(X\). The authors provide an example of a compact Hausdorff space of size \(2^c\) which is \(T_1\)-complementary to itself. They show that the existence of a compact Hausdorff space of size \(c\) that is \(T_1\)-complementary to itself is both consistent with and indepentdent of ZFC. On the other hand, a countably compact Tikhonov space of size \(c\) which is \(T_1\)-complementary to itself and a compact Hausdorff space of size \(c\) which is \(T_1\)-complementary to a countably compact Tikhonov space are constructed, both in ZFC. The results of the authors provide complete solutions to Problems 160 and 161 posed by S. Watson in [J. van Mill and G. M. Reed (ed.); Open Problems in Topology, North-Holland (1990; Zbl 0718.54001)].

MSC:
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54A35 Consistency and independence results in general topology
54G20 Counterexamples in general topology
54D30 Compactness
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