A compact Hausdorff topology that is a \(T_1\)-complement of itself.

*(English)*Zbl 1020.54002Two 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)].

Reviewer: Maximilian Ganster (Graz)

##### MSC:

54A10 | Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) |

54D20 | Noncompact covering properties (paracompact, LindelĂ¶f, 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 |