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Let $W$ be a Coxeter group and let $\Phi^+$ be its positive roots. A subset$B$ of $\Phi^+$ is called biclosed if, whenever we have roots $\alpha$, $\beta$and $\gamma$ with $\gamma \in \mathbb{R}_{>0} \alpha + \mathbb{R}_{>0} \beta$,if $\alpha$ and $\beta \in B$ then $\gamma \in B$ and, if $\alpha$ and $\beta\not\in B$, then $\gamma \not\in B$. The finite biclosed sets are the inversionsets of the elements of $W$, and the containment between finite inversion setsis the weak order on $W$. Matthew Dyer suggested studying the poset of allbiclosed subsets of $\Phi^+$, ordered by containment, and conjectured that itis a complete lattice. As progress towards Dyer's conjecture, we classify allbiclosed sets in the affine root systems. We provide both a type uniformdescription, and concrete models in the classical types $\widetilde{A}$,$\widetilde{B}$, $\widetilde{C}$, $\widetilde{D}$. We use our models to provethat biclosed sets form a complete lattice in types $\widetilde{A}$ and$\widetilde{C}$.

Combinatorial descriptions of biclosed sets in affine type