Partial Orders of Bijectively Related or Homeomorphic Topologies

Abstract

Topologies $\tau , \sigma \in \mathop{{\mathrm{Top}}}\nolimits _X$ are bijectively related, in notation $\tau \sim \sigma$, if there are continuous bijections $f: (X, \tau )\rightarrow (X, \sigma )$ and $g: (X, \sigma)\rightarrow (X, \tau)$. Defining $[\tau ]_{\cong}=\{ \sigma \in \mathop{{\mathrm{Top}}}\nolimits _X : \sigma \cong \tau\}$ and $[\tau ]_{\sim }=\{ \sigma \in \mathop{{\mathrm{Top}}}\nolimits _X : \sigma \sim \tau\}$ we show that for each infinite 1-homogeneous linear order ${\mathbb L}$ there is a topology $\tau \in \mathop{{\mathrm{Top}}}\nolimits _{|L|}$ such that: (a) $\langle [\tau ]_{\cong}, \subset \rangle \cong \dot{\bigcup}_{2^{|L|}}{\mathbb L}$ (the disjoint union of $2^{|L|}$-many copies of ${\mathbb L}$); so, each maximal chain in $[\tau ]_{\cong}$ is isomorphic to ${\mathbb L}$; (b) $\langle [\tau ]_{\sim}, \subset \rangle\cong \dot{\bigcup}_{2^{|L|}}\widetilde{{\mathbb L}}$, where $\widetilde{{\mathbb L}}$ is the Dedekind completion of ${\mathbb L}$; thus, each maximal chain in $[\tau ]_{\sim}$ is isomorphic to $\widetilde{{\mathbb L}}$. If, in addition, the linear order ${\mathbb L}$ is Dedekind complete, then the topology $\tau$ is weakly reversible, non-reversible and $\langle [\tau ]_{\sim}, \subset \rangle=\langle [\tau ]_{\cong}, \subset \rangle\cong \dot{\bigcup}_{2^{|L|}}{\mathbb L}$.