Marasinghe, M.M.S.H.K.Amarasinghe, A.K.2025-11-182025-11-182021-10-29Proceedings of the Postgraduate Institute of Science Research Congress (RESCON) -2021, University of Peradeniya, P 71978-955-8787-09-0https://ir.lib.pdn.ac.lk/handle/20.500.14444/6747In Riemannian geometry, introducing geometric concepts to smooth manifolds is done via selecting an appropriate Riemannian metric. We define 𝑅≥0 (𝑅 2 ) to be the space of all complete Riemannian metrics of non-negative curvature on the plane. The Lie group 𝐷𝑖𝑓𝑓(𝑅² ) of all self diffeomorphisms onto 𝑅² acts on 𝑅≥0 (𝑅² ) by pulling back metrics. Denote the moduli space of all complete Riemannian metrics of non-negative curvature on the plane by 𝑀≥0 (𝑅² ), it is the quotient space of 𝑅≥0 (𝑅² ) by the 𝐷𝑖𝑓𝑓(𝑅² ) action via pullback. The moduli space 𝑀≥0 (𝑅² ) is not a manifold since different Riemannian metrics may have isometry groups of different dimensions. A topological space 𝑋 is said to be weakly infinite-dimensional if for every family {(𝐴𝑖 , 𝐵𝑖 ):𝑖 ∈ 𝑁} of pairs of disjoint closed subsets of 𝑋, there exist separators 𝐷𝑖 between 𝐴𝑖 and 𝐵𝑖 such that ⋂𝑖=1 ∞ 𝐷𝑖 = ∅. The connectedness properties of the space 𝑅≥0 (𝑅² ) and 𝑀≥0 (𝑅² ) were first studied by Belegradek and Hu, and they proved that the complement of every finitedimensional subset of the space 𝑅≥0 (𝑅² ) is continuum-connected. It was later proved that the complement of every closed, finite-dimensional subset of 𝑅≥0 (𝑅² ) is path-connected and that the complement of a subset of 𝑀≥0 (𝑅² ) is path-connected if the subset is countable, or it is closed, metrisable and finite-dimensional. The results for 𝑅≥0 (𝑅² ) were generalised to show that complement of every closed, weakly infinite-dimensional subset of 𝑅≥0 (𝑅² ) is pathconnected. Further, a partial generalisation on 𝑀≥0 (𝑅² ) was obtained to prove that the complement of a closed Hausdorff space with Haver’s property 𝐶 of 𝑀≥0 (𝑅² ) is pathconnected. In this research, we prove that the complement of every closed Hausdorff weakly infinite-dimensional subset of 𝑀≥0 (𝑅² ) is path-connected, with an argument using a dimension theoretic argument on the dimensionality of a paracompact preimage of a fully closed map onto a weakly infinite-dimensional space. With this result, we conclude the series of theorems of connectedness properties of 𝑅≥0 (𝑅² ) and 𝑀≥0 (𝑅² ).en-USModuli spaceRiemannian metricsWeakly infinite-dimensionalArticle