Isomorphisms between the induced topologies on countable spaces A topological space $T$ is called canonical if all induced topologies on its infinite subsets are homeomorphic to $T$.
Is it true that every topology on an infinite countable set has a subset upon which the induced topology is canonical?
Thanks in advance.
 A: The following paper by Ginsburg and Sands offers a beautiful solution to this question.
https://www.jstor.org/stable/2320588?seq=1#metadata_info_tab_contents
Edit: The statement is correct, and this theorem allows one to prove that the only canonical topological spaces are those with

*

*Discrete topology.

*Indiscrete topology.

*Cofinite topology.

*Initial segment topology.

*Final segment topology.

A trivial consequence is that any infinite topological space contains (at least) one of the above topological subspaces.
The proof essentially classifies the types of canonical subspaces that a space $T$ may contain by considering first the sets $cl\{a\}$ for each point $a\in T$. The induced topology on each $cl\{a\}$ is the indiscrete topology, hence if any of the $cl\{a\}$ are infinite we are done. Otherwise, all $cl\{a\}$ are finite and in particular we have infinitely many of them so we may replace the space $T$ by a space $T_0$ by taking a single member of the closure of each element, the new space $T_0$ has the separation property, namely that for all $a,b\in T_0$, $cl\{a\} \neq cl\{b\}$. One may define a partial ordering of the space by $a\le b$ if $a\in cl\{b\}$ and use a classification theorem of infinite graphs by Ramsey and Whaley saying that an infinite graph must contain an infinite chain, an infinite antichain, or one other thing that is harder to describe.
If $T_0$ contains an infinite chain, i.e. it has elements $a_1 < a_2 < ...$, then if $A$ is a non-empty open subset of $T_0$ intersecting the chain, then it has a minimal element, say $a_k$, and since an open set containing $a_k$ must contain $a_n$ for all $n\ge k$, by considering the subspace induced by the chain we obtain a subspace with the final segment topology. One continues by case analysis in the same spirit.
