On the Inverse Limits of T 0

We show that if X is a locally compact, paracompact and Hausdorff space, then X can be realised as the subspace of all maximal points of the inverse limit of an inverse system of partial orders with an appropriate topology (equivalently T0-Alexandroff spaces). Then, the space X is homeomorphic to a deformation retract of that limit. Moreover, we extend results obtained by Clader and Thibault and show that if K is a simplicial complex, then its realisation |K| can be obtained as the subspace of all maximals of the limit of an inverse system of T0-Alexandroff spaces such that each of them is weakly homotopy equivalent to |K|. Moreover, if K is locally-finite-dimensional and |K| is considered with the metric topology, then this inverse system can be replaced by an inverse sequence.