Exterior space

In mathematics, the notion of externology in a topological space "X" generalizes the basic properties of the family
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 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.
Definition.
Let (X,τ) be a topological space. An externology on (X,τ) is a non-empty collection ε of open subsets satisfying:
An exterior space (X,τ,ε) consists of a topological space (X,τ) together with an externology ε. An open E which is in ε is said to be an exterior-open subset. A map f:(X,τ,ε) → (X',τ',ε') is said to be an exterior map if it is continuous and f−1(E) ∈ ε, for all E ∈ ε'.
The category of exterior spaces and exterior maps will be denoted by E. It is remarkable that E is a complete and cocomplete category.