Counting linearly ordered spaces

Gerald Kuba¹

¹ Institute of Mathematics University of Natural Resources and Life Sciences 1180 Wien, Austria

Colloquium Mathematicum, Tome 135 (2014) no. 1, pp. 1-14.

Résumé

For a transfinite cardinal $\kappa $ and $i\in \{0,1,2\}$ let ${\cal L}_i(\kappa )$ be the class of all linearly ordered spaces $X$ of size $\kappa $ such that $X$ is {totally disconnected} when $i=0$, the topology of $X$ is generated by a {dense} linear ordering of $X$ when $i=1$, and $X$ is {compact} when $i=2$. Thus every space in ${\cal L}_1(\kappa )\cap {\cal L}_2(\kappa )$ is connected and hence ${\cal L}_1(\kappa )\cap {\cal L}_2(\kappa )=\emptyset $ if $\kappa 2^{\aleph _0}$, and ${\cal L}_0(\kappa )\cap {\cal L}_1(\kappa )\cap {\cal L}_2(\kappa )=\emptyset $ for arbitrary $\kappa $. All spaces in ${\cal L}_1(\aleph _0)$ are homeomorphic, while ${\cal L}_2(\aleph _0)$ contains precisely $\aleph _1$ spaces up to homeomorphism. The class ${\cal L}_1(\kappa )\cap {\cal L}_2(\kappa )$ contains precisely $2^\kappa $ spaces up to homeomorphism for every $\kappa \geq 2^{\aleph _0}$. Our main results are explicit constructions which prove that both classes ${\cal L}_0(\kappa )\cap {\cal L}_1(\kappa )$ and ${\cal L}_0(\kappa )\cap {\cal L}_2(\kappa )$ contain precisely $2^\kappa $ spaces up to homeomorphism for every $\kappa >\aleph _0$. Moreover, for any $\kappa $ we investigate the variety of second countable spaces in the class ${\cal L}_0(\kappa )\cap {\cal L}_1(\kappa )$ and the variety of first countable spaces of arbitrary weight in the class ${\cal L}_2(\kappa )$.

Détail

DOI : 10.4064/cm135-1-1

Keywords: transfinite cardinal kappa cal kappa class linearly ordered spaces size kappa totally disconnected topology generated dense linear ordering compact every space cal kappa cap cal kappa connected hence cal kappa cap cal kappa emptyset kappa aleph cal kappa cap cal kappa cap cal kappa emptyset arbitrary kappa spaces cal aleph homeomorphic while cal aleph contains precisely aleph spaces homeomorphism class cal kappa cap cal kappa contains precisely kappa spaces homeomorphism every kappa geq aleph main results explicit constructions which prove classes cal kappa cap cal kappa cal kappa cap cal kappa contain precisely kappa spaces homeomorphism every kappa aleph moreover kappa investigate variety second countable spaces class cal kappa cap cal kappa variety first countable spaces arbitrary weight class cal kappa