We prove that any compact semi-algebraic set is homeomorphic to the solution space of some art gallery problem. Previous works have established similar universality theorems, but holding only up to homotopy equivalence, rather than homeomorphism, and prior to this work, the existence of art galleries even for simple spaces such as the M\"obius strip or the three-holed torus were unknown. Our construction relies on an elegant and versatile gadget to copy guard positions with minimal overhead. It is simpler than previous constructions, consisting of a single rectangular room with convex slits cut out from the edges. We show that both the orientable and non-orientable surfaces of genus $n$ admit galleries with only $O(n)$ vertices.

翻译：我们证明了任何紧半代数集同胚于某个艺术画廊问题的解空间。先前的研究已建立类似的普遍性定理，但仅适用于同伦等价而非同胚，且在本工作之前，即使对于莫比乌斯带或三孔环面等简单空间，艺术画廊的存在性也未知。我们的构造依赖于一种优雅且通用的装置，能以最小开销复制守卫位置。该构造比先前的工作更简洁，仅包含一个矩形房间及从边缘切出的凸缝。我们证明，无论可定向还是不可定向的亏格为$n$的曲面，都存在仅含$O(n)$个顶点的画廊。