ContinuumFromS

ContinuumFromS is a theoretical construction in topology that, given a subset S of a topological space X, produces a compact connected metric space C(S) known as the continuum from S. The construction is used to study how the structure of S influences the topology of the resulting continuum and is explored within the field of continuum theory.

Definition and general construction: For a compact subset S of a locally compact, second-countable Hausdorff space, one common realization forms a family of arcs attached to a fixed base point and connected to each s in S at the point s. The far endpoints of these arcs are then identified to a single apex, and the resulting space is closed to obtain a compact, connected space. In specific implementations, S is embedded into C(S) via s ↦ the attachment point corresponding to s; varying the attachment scheme yields different continua associated with the same S.

Properties: If S is compact, C(S) is a continuum (compact, connected, metrizable). The embedding of S into C(S) preserves many topological features of S when S is locally connected, and the construction is functorial with respect to inclusions S ⊆ T, giving a continuous map C(S) → C(T). The dimension and local connectedness of C(S) reflect properties of S under suitable regularity conditions.

Examples: When S consists of finitely many points, C(S) resembles a finite-armed graph with a common center. If S is a Cantor set, C(S) can yield a highly branching, dendroid-like continuum that encodes the Cantor structure in its topology. Different attachment choices can produce non-homeomorphic continua from the same S.

Applications: The construction is used in descriptive set theory, continuum theory, and dynamical systems to model fill-in procedures and to study how embeddings of S behave within different continua. See also continuum theory, cone constructions, and topological graph concepts.