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A code $$\mathcal{C} \subset 2^{[n]}$$ is open convex if there is a collection $$\mathcal{U}$$ of open convex subsets of $$\mathbb{R}^d$$ (for some $$d \geq 1$$) so that $$\mathrm{code}(\mathcal{U}) = \mathcal{C}$$. If $$\mathcal{U}$$ is instead a collection of closed convex subsets of $$\mathbb{R}^d$$ and $$\mathrm{code}(\mathcal{U}) = \mathcal{C}$$, then $$\mathcal{C}$$ is closed convex.
It's easy to see (but technical to prove) that every open convex code is also closed convex (shrink each of the open sets by a small enough $$\varepsilon$$ so that the intersection pattern is not altered, then take closure of the new sets), but surprisingly there are closed convex codes which are not open convex. An example of such a code is provided in Theorem 3.1 of Obstructions to convexity in neural codes by Lienkaemper, Shiu, and Woodstock. (In that paper, the authors use the term convex to convey what I have defined above as open convex.) There, they show that the code $\mathcal{C} = \{2345, 123, 134, 145, 13, 14, 23, 34, 45, 3, 4,\varnothing\}$ is not open convex. However, there is a collection $$\mathcal{U}$$ of closed convex subsets of $$\mathbb{R}^2$$ for which $$\mathrm{code}(\mathcal{U}) = \mathcal{C}$$, and one such collection is provided in a forthcoming paper by Giusti, Itskov, and myself.