Partial Orders of Bijectively Related or Homeomorphic Topologies

TL;DR

This paper explores the structure of topologies on infinite linear orders, establishing relationships between bijective relatedness and homeomorphism classes, and characterizing their order-theoretic properties.

Contribution

It constructs specific topologies on infinite linear orders that reveal detailed order structures of related and homeomorphic topologies, including their maximal chains and reversibility properties.

Findings

Maximal chains in homeomorphism classes are isomorphic to the linear order or its Dedekind completion.

The topology constructed is weakly reversible and non-reversible when the order is Dedekind complete.

The order structures of related and homeomorphic topologies are characterized as disjoint unions of copies of the linear order and its completion.

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 $\langle [\tau ]_{\sim}, \subset \rangle=\langle [\tau ]_{\cong}, \subset \rangle\cong \dot{\bigcup}_{2^{|L|}}{\mathbb L}$.