On the remainder of the semialgebraic Stone–Cěch compactification of a semialgebraic set

Abstract

In this work we analyze some topological properties of the remainder ∂M:=βs⁎M∖M of the semialgebraic Stone–Cěch compactification βs⁎M of a semialgebraic set M⊂Rm in order to ‘distinguish’ its points from those of M. To that end we prove that the set of points of βs⁎M that admit a metrizable neighborhood in βs⁎M equals Mlc∪(Clβs⁎M(M‾≤1)∖M‾≤1) where Mlc 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 ∂ˆM 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∖∂ˆM and ∂ˆM∖∂˜M are also dense subsets of ∂M. It holds moreover that all the points of ∂ˆM have countable systems of neighborhoods in βs⁎M.