Product of topological spaces and product of corresponding locales

Let $$\Omega : \mathbf{Top} \to \mathbf{Loc}$$ the functor that takes a topological space to its locale of opens. For a locale morphism $$f$$, write $$f^*$$ for the correspoding morphism in $$\mathbf{Frm}$$, the category of frames and frame homomorphisms. Let $$\times_t$$ denote the product in $$\mathbf{Top}$$ and $$\times_{\ell}$$ the product in $$\mathbf{Loc}$$ (= the coproduct in $$\mathbf{Frm}$$).

Let $$X$$ and $$Y$$ be topological spaces. We can describe $$\Omega X \times_{\ell} \Omega Y$$ as the locale generated by pairs $$[a, b]$$, where $$a \in X$$ and $$b \in Y$$, subject to the relations \begin{align*} &[\top, \top] = \top \\ &[a, b] \wedge [a', b'] = [a \wedge a', b \wedge b'] \\ &[a, \bigvee S] = \bigvee \{ [a, b] \mid b \in S \} \\ &[\bigvee T, b] = \bigvee \{ [a, b] \mid a \in T \} \end{align*} where $$T \subseteq X$$ and $$S \subseteq Y$$.

Define $$f : \Omega(X \times_t Y) \to \Omega X \times_{\ell} \Omega Y$$ by $$f^*([a, b]) = a \times b,$$ where $$a \times b$$ denotes the open rectangle in $$X \times_t Y$$. Then it is easy to verify that $$f^*$$ is a frame homomorphism, hence $$f$$ is a locale morphism. Furthermore, $$f^*$$ is onto, since the topology of $$X \times_t Y$$ is generated by a base consisting of open rectangles $$a \times b$$.

Question: Is $$f$$ an isomorphism? In other words, is $$f^*$$ injective?

• I hear it's folklore that this isn't an isomorphism for $X = Y = \mathbb{Q}$. – Qiaochu Yuan Sep 14 '19 at 0:53

Your map $$f$$ is known to be an injective dense localic map. See, for example, Proposition 4.2.2 in [1]. In general, it isn't an isomorphism. The reason for this is that $$\Omega(X \times_t Y)$$ is quotiented by more equations than $$\Omega(X) \times_\ell \Omega(Y)$$ is.

We can think of $$\Omega(X \times_t Y)$$ as if it was generated from the same set of generators as $$\Omega(X) \times_\ell \Omega(Y)$$ but quotiented with respect to the following equation:

$$\bigvee_i [U_i, V_i] \sim \bigvee_j [U'_j, V'_j] \quad\text{ iff }\quad \bigcup_i (U_i\times V_i) = \bigcup_j (U'_j \times V'_j).$$

The right-hand size is computed with respect to the points of $$X\times Y$$. It is easy to see that $$\sim$$ is a stronger relation than the one that defines $$\Omega(X) \times_\ell \Omega(Y)$$.

It is also known that, if $$X$$ and $$Y$$ are sober locally compact, then $$f$$ is an isomorphism. (Proposition 4.4.1. in [1]).

In many ways, the frame-theoretic product behaves the same way as the tensor product of abelian groups. For details see [1,2].

References

1. Jorge Picado, Aleš Pultr. Notes on the product of locales. Mathematica Slovaca 65.2 (2015): 247-264. PDF
2. Jorge Picado and Aleš Pultr. Frames and Locales: topology without points. Springer Science & Business Media, 2011.