This question basically follows [this earlier question of mine](https://mathoverflow.net/q/353184/8133) but shifting from standard systems of nonstandard models of $PA$ to $\omega$-models of $RCA\_0$. For $X$ a Turing ideal we get the map $c\_X$ on $2^\omega$ given by $c\_X(x)=[b[x]]\cap X$ where $b$ is some computable bijection $\omega\cong 2^{<\omega}$; in the language of the linked question, this is a **closed set pattern**.


I'm trying to understand how much information this construction loses in the sense of second-order arithmetic. For $X$ a Turing ideal we *also* get an $\omega$-model of $RCA\_0$ - **which I'll freely conflate with $X$ itself** - and I'm interested in what axioms of second-order arithmetic we can detect topologically.


(Below, $\sim$ denotes "difference by homeomorphism" - for $c,d$ closed set patterns on $\mathcal{X},\mathcal{Y}$ we write $c\sim d$ iff for some $H:\mathcal{X}\cong\mathcal{Y}$ we have $a\in c(b)\leftrightarrow H(a)\in d(H(b))$ for all $a,b\in\mathcal{X}$.)


Say that a theory of second-order arithmetic is *topologically detectable* if there is some $\sim$-respecting property of closed set patterns which holds of $c\_X$ iff $X\models A$ for each Turing ideal $X$. For example:


* $WKL\_0$ is topologically detectable via "$c\_X^{-1}(\emptyset)$ is open."
* $ACA\_0$ is topologically detectable in at least two ways *(after adding $WKL\_0$)*: via "$ran(c\_X)$ is closed under (single) Cantor-Bendixson derivatives" and "for each $x\in X$ the set $\{y\in X: c(y)\supseteq c(x)\}$ is closed."


My question is essentially whether any other "weak combinatorial principles" are topologically detectable:



> 
> Is there some "reasonably natural" $A$ with $ACA\_0\models A$ and $WKL\_0\not\models\_\omega A$ such that $A$ (or at least $WKL\_0+A$) is topologically detectable?
> 
> 
> 


*(Here $\models\_\omega$ is the restriction of $\models$ to $\omega$-models, which is needed to rule out e.g. $I\Sigma\_{17}$. Note that if $A$ is topologically detectable then so is $WKL\_0+A$, but the converse isn't obvious to me.)* 


The most tempting candidate is of course $RT^2\_2$, but there are plenty of others.



Detecting comprehension topologically
Statements about existence of $\omega$-models can be topologically detected.


Specifically, fix $X$ a Turing ideal. For $t\in X$ say that $t$ *enumerates a family of sets* if:


* Exactly one $p\in c\_X(t)\cap X$ has $c\_X(p)=X$.
* For every other $q\in c\_X(t)$ we have $c\_X(q)=\{a\}$ for some $a\in y$.
* For each $a\in y$ there is exactly one $q\in c\_X(t)$ with $c\_X(q)=\{a\}$.


In such a case we say $t$ enumerates the family $$X\_t:=\{q\in X: \exists a\in c\_X(t)(c\_X(a)=\{q\})\},$$ and we can talk about the induced closed set pattern coming from $X\_t$. It's not hard to see$^\*$ that every sequence of sets in $X$ (that is, the whole sequence is in $X$) corresponds to such an $X\_t$, and this means:



> 
> If $A$ is a topologically detectable sentence, so is the statement $O\_A$ = "Every real is contained in an $\omega$-model of $A$."
> 
> 
> 


Taking $A=WKL\_0$ then gives an affirmative answer to the question. Of course $RCA\_0+O\_{WKL\_0}\vdash WKL\_0$: if $X$ is an $\omega$-model of $RCA\_0$ and $T$ is an infinite binary tree in $X$, then any $\omega$-model of $WKL\_0$ containing $T$ also thinks $T$ is an infinite binary tree - and being a path through a tree is absolute between $\omega$-models. (More generally, we have $$RCA\_0+ O\_\varphi\vdash\varphi$$ for *every* $\varphi\in\Pi^1\_2$.) 


Meanwhile, $RCA\_0+O\_{RCA\_0}\vdash WKL\_0$, since from a coded $\omega$-model of $RCA\_0$ we can whip up a $DNR\_2$ function (and this relativizes). So this approach **does not** produce an example of a topologically detectable sentence incomparable with $WKL\_0$.




---


$^\*$Specifically, given a sequence of reals $F=(f\_i)\_{i\in\omega}$ let $s\_i$ be the natural code for the tree $$\{\sigma\in 2^{<\omega}: \vert\sigma\vert<i\mbox{ or }\sigma\prec f\}.$$ The closure of $\{s\_i: i\in\omega\}$ is the set of paths through a pruned tree $T$ - which is computable in $F$ - and $[T]$ has only one "extra" path, corresponding to $2^{<\omega}$.


