Let $X$ be a topological space and $x \in X$ a point. Let $\Omega$ be the set of open sets (viꝫ. the topology) of $X$, and $\Omega_x$ the set of *germs* around $x$ of open sets, that is, $\Omega_x = \Omega/{\sim}$ where $U\sim V$ means there exists a neighborhood $W$ of $x$ such that $U\cap W = V\cap W$. (Note that $U$ and $V$ need not be neighborhoods of $x$.)

Clearly, $\Omega$, partially ordered by inclusion, is a distributive lattice, with join and meet operations being given by $\cup$ and $\cap$. This structure passes to the quotient in the obvious way, so $\Omega_x$ is a distributive lattice.

But $\Omega$ is more than a distributive lattice: it is a **frame** meaning that it has arbitrary joins (given by $\bigcup$), and that finite meets distribute over arbitrary joins. (A frame also has arbitrary meets, but they are not considered part of the structure and need not be preserved by homomorphisms.) Now this structure *does not* pass to the quotient $\Omega_x$ in the sense that the canonical surjection $\Omega \to \Omega_x$ does not, in general, define a frame structure on its target so that it is a frame homomorphism. (Indeed, it is easy to construct families $U_i$ and $V_i$ of open sets such that $U_i \sim V_i$ for every $i$ but $\bigcup_i U_i$ and $\bigcup_i V_i$ do not have the same germ, e.g., take $U_i = \varnothing$ in $X = \mathbb{R}$ and $V_i$ the complement of the closed interval $[-\frac{1}{i}, \frac{1}{i}]$ for $i\geq 1$ around $x=0$. But note that in this example, the join of the classes $[U_i] = [V_i]$ under $\sim$ still exists in $\Omega_x$, and is $[\varnothing]$. So this counterexample does not answer the following question:)

**Question:** Is $\Omega_x$ itself a frame (albeit not a quotient frame of $\Omega$)? In other words:

Does the sup of an arbitrary family of germs of open sets exist?

If so, do finite intersections (=meets) distribute over these arbitrary joins?

(In case the answer is negative, a counterexample with $X=\mathbb{R}$ would be most appreciated.)

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