Exterior space

In mathematics, the notion of externology in a topological space X generalizes the basic properties of the family

εXcc = {E ⊆ X : X\E is a closed compact subset of X}

of complements of the closed compact subspaces of X, which are used to construct its Alexandroff compactification. An externology permits to introduce a notion of end[1] point, to study the divergence of nets in terms of convergence to end points and it is a useful tool for the study and classification of some families of non compact topological spaces. It can also be used to approach a topological space as the limit of other topological spaces: the externologies are very useful when a compact metric space embedded in a Hilbert space is approached by its open neighbourhoods.

  1. ^ "proper homotopy theory in nLab". ncatlab.org.