Homeomorphic Non-Comparable Particular Point Topologies

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Let $S$ be a set with at least two elements.

Let $p, q \in S: p \ne q$.

Let $\tau_p$ and $\tau_q$ be the particular point topologies on $S$ by $p$ and $q$ respectively.

Then the topological spaces $T_p = \struct {S, \tau_p}$ and $T_q = \struct {S, \tau_q}$ are homeomorphic.

However, $\tau_p$ and $\tau_q$ are not comparable.


We can set up the mapping $\phi: S \to S$:

$\forall x \in S: \map \phi x = \begin{cases} q & : x = p \\ p & : x = q \\ x & : \text {otherwise} \end{cases}$

It is straightforward to show that $\phi$ is a homeomorphism.

However, we have, for example, that $\set q \notin \tau_p$ and $\set p \notin \tau_q$.

So neither $\tau_p$ nor $\tau_q$ are a subset of the other.

Hence by definition $\tau_p$ and $\tau_q$ are not comparable.