subcontinua

Subcontinua are nonempty compact connected subsets of a continuum. In topology, a continuum is a compact connected metric space, so a subcontinuum of a given X is a nonempty compact connected subset Y contained in X.

Basic properties: Because X is compact and metric, every subcontinuum is compact and closed in X, and hence itself a continuum. Intersections of nested families of subcontinua with nonempty intersection remain subcontinua, providing a way to approximate a subcontinuum by smaller pieces. The degenerate case of a subcontinuum is a single point, which is a compact connected subset of X.

Special cases and examples: If X is the unit interval [0,1], subcontinua are precisely the closed intervals [a,b] with 0 ≤ a ≤ b ≤ 1. In higher dimensions, subcontinua include many shapes such as arcs, simple closed curves, dendrites, and more complex continua contained in the ambient space.

Hyperspace of subcontinua: For a given continuum X, one studies the hyperspace C(X), the collection of all subcontinua of X, equipped with the Vietoris topology. When X is compact metric, C(X) itself is a compact metric space. The points of C(X) are the subcontinua of X, and the topology encodes how these sets can vary within X; degenerate subcontinua correspond to the singletons {x}.

Significance and scope: Subcontinua form a central object in continuum theory, providing a framework to decompose and analyze the structure of X through its connected, compact pieces. They play a role in dynamics, invariant sets, and the study of how a space can be built up from smaller continua.