In this work we analyze some topological properties of the remainder ∂M := β * s M \ M of the semialgebraic Stone-Cěch compactification β * s M of a semialgebraic set M ⊂ R m in order to 'distinguish' its points from those of M. To that end we prove that the set of points of β * s M that admit a metrizable neighborhood in β * s M equals M lc ∪ (Cl β * s M (M ≤1) \ M ≤1) where M lc is the largest locally compact dense subset of M and M ≤1 is the closure in M of the set of 1-dimensional points of M. In addition, we analyze the properties of the sets ∂M and ∂M of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder ∂M and that the differences ∂M \ ∂M and ∂M \ ∂M are also dense subsets of ∂M. It holds moreover that all the points of ∂M have countable systems of neighborhoods in β * s M .
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