A compact Hausdorff topology that is a $T_1$-complement of itself

Dmitri Shakhmatov¹ ; Michael Tkachenko²

¹ Department of Mathematical Sciences Faculty of Science Ehime University Matsuyama 790-8577, Japan
² Departamento de Matemáticas Universidad Autónoma Metropolitana Av. San Rafael Atlixco 186, Del. Iztapalapa C.P. 09340 México D.F., México

Fundamenta Mathematicae, Tome 175 (2002) no. 2, pp. 163-173.

Résumé

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).

Détail

DOI : 10.4064/fm175-2-6

Keywords: topologies tau tau set called complementary tau cap tau setminus subseteq finite cup emptyset tau cup tau subbase discrete topology topological spaces tau tau called complementary provided there exists bijection tau tau complementary topologies provide example compact hausdorff space size mathfrak which complementary itself mathfrak denotes cardinality continuum prove existence compact hausdorff space size mathfrak complementary itself consistent independent zfc other construct zfc countably compact tikhonov space size mathfrak which complementary itself compact hausdorff space size mathfrak which complementary countably compact tikhonov space examples have smallest possible size consistent zfc mathfrak smallest cardinality infinite set admitting hausdorff complementary topologies results provide complete solutions problems posed watson problems topology north holland