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