A compact Hausdorff topology that is a $T_1$-complement of itself
Volume 175 / 2002
Abstract
Topologies $\tau_1$ and $\tau_2$ on a set $X$ are called $T_1$-complementary if $\tau_1\cap\tau_2=\{X\setminus F: F\subseteq X$ is finite$\}\cup\{\emptyset\}$ and $\tau_1\cup\tau_2$ is a subbase for the discrete topology on $X$. Topological spaces $(X,\tau_X)$ and $(Y,\tau_Y)$ are called $T_1$-complementary provided that there exists a bijection $f: X\to Y$ such that $\tau_X$ and $\{f^{-1}(U):U\in\tau_Y\}$ are $T_1$-complementary topologies on $X$. We provide an example of a compact Hausdorff space of size $2^{\mathfrak c}$ which is $T_1$-complementary to itself (${\mathfrak c}$ denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff space of size ${\mathfrak c}$ that is $T_1$-complementary to itself is both consistent with and independent of ZFC. On the other hand, we construct in ZFC a countably compact Tikhonov space of size ${\mathfrak c}$ which is $T_1$-complementary to itself and a compact Hausdorff space of size ${\mathfrak c}$ which is $T_1$-complementary to a countably compact Tikhonov space. The last two examples have the smallest possible size: It is consistent with ZFC that ${\mathfrak c}$ is the smallest cardinality of an infinite set admitting two Hausdorff $T_1$-complementary topologies [8]. Our results provide complete solutions to Problems 160 and 161 (both posed by S.~Watson [14]) from Open Problems in Topology (North-Holland,~1990).
