Computational approximations of compact metric spaces

A combinatorial description of shape theory

“…) is an inverse sequence. This inverse sequence is similar to the Main Construction introduced in [3] and is isomorphic to every finite approximation (U 4 n (A n ), q n,n+1 ) of X in pro-HT op (see [11]).…”

“…We recall briefly the method obtained in [11] to reconstruct algebraic invariants of X. We say that a finite set A ⊂ X is an -approximation of X if for every x ∈ X there exists a ∈ A such that d(x, a) < , where is a positive real value.…”

“…For every n ∈ N we consider an n -approximation A n . The map q n,n+1 : [11]), where B(x, ) denotes the open ball of radius and center x. We say that the inverse sequence (U 4 n (A n ), q n,n+1 ) is a finite approximation of X.…”

“…Later in [12], E. Clader constructed for every simplicial complex X an inverse sequence of finite spaces satisfying that its inverse limit contains a homeomorphic copy of X which is a strong deformation retract. A generalization of this result to compact metric spaces was given in [33] and in [11], where computational aspects are studied and a method to reconstruct algebraic invariants is proposed. These methods are motivated somehow by the recent theory of topological data analysis [19,8], which has grown up very fast the last years (e.g.…”

“…The organization of this paper is as follows. In Section 2 we introduce the reformulation of the intrinsic description of shape theory [41] that was given in [2], we also recall the method to reconstruct algebraic invariants given in [11] and basic results in the theory of finite spaces. In Section 3 we describe our combinatorial approach of shape theory, define the notion of core for inverse sequences and get some shape invariants.…”