Partial Orders of Bijectively Related or Homeomorphic Topologies

Abstract

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

arXiv comment15 pages