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In mathematics, the notion of externology in a topological space ''X'' generalizes the basic properties of the family : ''ε''''X''cc = 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: * If E1, E2 ∈ ε, then E1 ∩ E2 ∈ ε; * if E ∈ ε, U ∈ τ and E ⊆ U, then U ∈ ε. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Exterior space」の詳細全文を読む スポンサード リンク
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