Glasnik Matematicki, Vol. 52, No. 2 (2017), 207-219.

ON THE INVERSE LIMITS OF T₀-ALEXANDROFF SPACES

Paweł Bilski

Abstract.   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.

2010 Mathematics Subject Classification.   06A06, 08B25, 55U10.

Key words and phrases.   Alexandroff space, inverse limit, locally compact space, paracompact space, partial order, simplicial complex, weak homotopy type.

DOI: 10.3336/gm.52.2.01

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