Counting linearly ordered spaces
Tom 135 / 2014
Streszczenie
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 ${\cal L}_2(\kappa )$.
