Subcompactsegment

Subcompactsegment is a geometric notion used to refer to a line segment that lies entirely within a subcompact subset of a larger metric or topological space. The term is not standard in mainstream textbooks but appears in discussions of how straight-line objects interact with compact regions in analysis and geometry.

Definition: Let X be a metric space, and let K ⊆ X be a subcompact subset (compact in the subspace topology). A subcompactsegment in K is the image γ([0,1]) of a continuous map γ:[0,1] → K. In Euclidean space, this coincides with a straight-line segment between two points of K that stays inside K; more generally, γ need not be linear, but its image remains contained in K.

Examples: In R^2, if K is the closed disk, any chord of the disk is a subcompactsegment. If K is a non-convex compact set, the straight line between two interior points may leave K, in which case that line is not a subcompactsegment.

Properties: A subcompactsegment is compact as the continuous image of [0,1], and it is connected. If K is convex, every line segment with endpoints in K is a subcompactsegment.

Applications: Used in the study of how linear or near-linear geometric objects interact with compact regions, with relevance to path planning within bounded obstacles and to convergence questions in approximation schemes.

See also: compact subset, subspace, convex set, geodesic.