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 be a topological space. An externology on is a non-empty collection ε of open subsets satisfying:
An exterior space consists of a topological space together with an externology ε. An open E which is in ε is said to be an exterior-open subset. A map f: → is said to be an exterior map if it is continuous and f−1 ∈ 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.

Some examples of exterior spaces