covering vs exterior

covering

• That which covers or conceals; a cover; something spread or laid over or wrapped about another. 

• Action of the verb to cover. 

• A continuous map onto a topological space X, 𝜋:E→X, such that there exists a discrete space D and for every x∈X an open neighborhood U⊂X, such that 𝜋⁻¹(U)= displaystyle bigsqcup _(d∈D)V_d and 𝜋|_(V_d):V_d→U is a homeomorphism for every d∈D. 

exterior

• The outside part, parts or surface of something. 

• Foreign lands. 

• Being from outside a country; foreign. 

• Outdoor. 

• Relating to the outside parts or surface of something.