The motivation for this question is a bit convoluted, so in the interests of conciseness I'm just asking it as a curiosity (and I do find it interesting on its own); if anyone is interested, feel free to email me.

I'm playing around with various notions of describing topological spaces, and I've found the following fun:

Let Baire space $\mathcal{N}$ be $\mathbb{N}^\mathbb{N}$ with the usual topology. Say that a space $\mathcal{X}=(X,\tau)$ is pictorial if there are $A,B\subseteq\mathcal{N}$ arbitrary and $R\subseteq\mathcal{N}^2$ open such that $$\{\{a\in A: aRb\}: b\in B\}$$ generates a topology $\sigma$ on $A$ with $\mathcal{X}\cong(A,\sigma)$.

Of course, we could replace $\mathcal{N}$ with an arbitrary space, or even (distinguishing between "point part" and "set part") an arbitrary pair of spaces, but at present I don't see what that added generality gives us.

My question is simply:

What spaces are pictorial?

Or even better:

Are some good sources on this topic/type of problem?

I haven't been able to find any, but I'm not familiar with the relevant literature.

There are a couple easy observations:

  • All pictorial spaces are separable. (And truly trivially, all pictorial spaces have at most continuum many points and a base of size at most continuum.)

  • Subspaces, countable products, and $T_0$-ifications of pictorial spaces are pictorial.

  • There are Hausdorff non-first-countable pictorial spaces, and there are $T_1$ pictorial spaces such that no point is characterized uniquely by countably many open sets; on the other hand, if $(X,\tau)$ is Hausdorff and pictorial, then for every point $p\in X$ there is a countable family $(U_i)_{i\in\mathbb{N}}$ of open sets characterizing $p$ completely. (A point $p$ is characterized uniquely by countably many open sets if there is a sequence $(U_i)_{i\in\mathbb{N}}$ of open sets such that for all points $q$ we have $\{i: q\in U_i\}=\{i: p\in U_i\}\iff p=q$).

However, I have no idea how to prove any nontrivial results about pictorial spaces, the main issue being the difficulty of proving any negative results. For example, I don't know whether the continuous image of a pictorial space need be pictorial: removing points from the $B$-part is tempting at first but the pullback of a subbase of the target may not look anything like the subbase witnessing the pictoriality of the starting space.

  • $\begingroup$ What is the Hausdorff non-first-countable example? $\endgroup$ – James Hanson Aug 7 '18 at 14:43
  • $\begingroup$ @JamesHanson The space of continuous functions $[0,1]\rightarrow [0,1]$ with the pointwise-convergence topology. On the $A$-side we use appropriate codes for continuous functions (= sequences specifying which rational open balls map into which other open balls), and on the $B$-side we use appropriate codes for sets of the form "the set of functions whose values on real $r$ are between rationals $q_0$ and $q_1$." If a continuous function is in an open set, this is determined by some finite initial segment of a code for each. $\endgroup$ – Noah Schweber Aug 7 '18 at 20:37
  • $\begingroup$ The other example is a variation: extend attention to the analogous space of continuous functions minus finitely many points. $\endgroup$ – Noah Schweber Aug 7 '18 at 20:38
  • $\begingroup$ Minus finitely many points from what? Also what does the second example accomplish? $\endgroup$ – James Hanson Aug 7 '18 at 23:54
  • 1
    $\begingroup$ @JamesHanson Yes, that's true. (But it ruins part of the fun, which is drawing open sets and then figuring out what spaces they describe!) $\endgroup$ – Noah Schweber Aug 8 '18 at 1:10

I have two things to offer, the first of which could help with getting better characterizations, the latter should give ample of examples.

Since I am not aware of standard terminology, call $(X,\tau')$ a topological weakening of $(X,\tau)$ if $\tau' \subseteq \tau$.

Theorem: A space is pictorial iff it is homeomorphic to a topological weakening of a subspace of $\mathcal{N}$.

For the forward implication, note that for every $b \in B$ the set $\{a \mid aRb\}$ is open in the subspace topology that $A$ inherits from $\mathcal{N}$, hence the entire topology we induce on $A$ is a subset of its subspace topology.

For the converse implication, take the standard coding of open subsets of $\mathcal{N}$ over $\mathcal{N}$, and note that p is contained in the open set coded by q is an open relation on $\mathcal{N}$. So to see that any topological weakening $(A,\tau)$ of a subspace $\mathbf{A}$ is pictorial, just take the underlying set $A$, and for $B$ the set of all codes for extensions of opens in $\tau$.

We get plenty of preservation results as corollaries, namely everything that commutes with weakening of the topology and preserves subspaces of $\mathcal{N}$.

Theorem: Every sequential $\mathrm{QCB}_0$ space[1] is pictorial.

The sequential $\mathrm{QCB}_0$-spaces are precisely those topological spaces that arise as represented spaces in computable analysis. A represented space is just a pair $(X,\rho)$, where $\rho : \subseteq \mathcal{N} \to X$ is a partial surjection onto the set $X$.

Whenver $\mathbf{X}$ is a represented space, then so is the space $\mathcal{O}(\mathbf{X})$ of its open subsets[2]. The relationship $x \in U$ is open on $\mathbf{X} \times \mathcal{O}(\mathbf{X})$, which can be pulled back to the level of $\mathcal{N}$. Now just choose a single representative from each $\rho_\mathbf{X}^{-1}(\{x\})$ for $x \in \mathbf{X}$ to construct $A$. For $B$, we can just take the domain of the representation of $\mathcal{O}(\mathbf{X})$.

The second observation accounts for all sequential separable spaces "I have come across naturally" as being pictorial.

[1] $\mathrm{QCB}_0$ spaces are $T_0$ quotients of countably-based spaces. See https://www.sciencedirect.com/science/article/pii/S0304397501001098 for more.

[2] A subset $U \subseteq \mathbf{X}$ of a represented space $(X,\delta)$ is open, if $\delta^{-1}(U)$ is open in $dom(\delta)$.

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  • $\begingroup$ @WlodAA Either I am misunderstanding you, or vice versa. We are moving to a subspace before taking the topological weakening (we could do it in the other order, too). I agree that taking the subspace is probably necessary, but it is mentioned in the theorem. $\endgroup$ – Arno Aug 13 '18 at 7:57
  • $\begingroup$ Arno, I apologize. I had a total blackout, sorry. (Yes, there was sub to which I was somehow blind). $\endgroup$ – Wlod AA Aug 13 '18 at 9:53
  • $\begingroup$ A couple belated comments: (i) What is the definition of sequential QCB$_0$ space? (ii) I'm new to represented spaces; I assume that if $(X,\rho)$ is a represented space, then the opens of $\mathbb{X}$ are precisely the sets whose $\rho$-preimage is open in $\mathcal{N}$? $\endgroup$ – Noah Schweber Oct 9 '18 at 14:59
  • $\begingroup$ @NoahSchweber I've added responses. $\endgroup$ – Arno Oct 10 '18 at 14:04
  • $\begingroup$ Thanks, I've accepted since I think this comes as close to a fully-satisfying answer as can be expected. Incidentally, do you know if there is a non-QCB$_0$ space which is pictorial? I suspect there is, but I don't see how to cook one up. $\endgroup$ – Noah Schweber Oct 13 '18 at 23:28

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