Problem 5. Prove that for any finite X C R", Rips(X, r) C Čech(X, 2r).

To prove that for any finite X⊂ℝn, Rips(X,r)⊆check{C}(X,2r), we need to show that any simplex in the…

Answered: Prove that for any finite X C R¹,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Problem 5. Prove that for any finite X C R", Rips(X, r) C Čech(X, 2r).

Expert Solution

Step 1

To prove that for any finite $X \subset ℝ^{n}$ , $R i p s (X, r) \subseteq check {C} (X, 2 r)$ , we need to show that any simplex in the Rips complex with the diameter at most $r$ is also in the Čech complex with a radius $2 r$ .

Rips Complex:

The Rips complex is a geometric construction in topology that encodes the topological structure of a set of points in Euclidean space. Given a finite set of points in Euclidean space, the Rips complex is formed by considering all possible subsets of the points whose pairwise distances are less than or equal to a certain fixed radius. Specifically, the Rips complex is a simplicial complex that contains all simplices spanned by these subsets of points.

More formally, given a set of points X in Euclidean space, the Rips complex of X at scale r is defined as follows: the vertices of the complex are the points of X, and a set of k+1 vertices forms a k-simplex in the complex if the pairwise distances between these points are all less than or equal to r.

The Rips complex is an example of a topological space that is constructed from a metric space, and it can be used to study the topological properties of the original point set. The Rips complex is also closely related to other constructions in topological data analysis, such as the Čech complex and the alpha complex.

Čech complex:

The Čech complex is a method for constructing a simplicial complex from a topological space. Given a topological space X and a cover $U_{i}$ of X by open sets, the Čech complex $C (X, U_{i})$ is defined as follows:

The vertices of the complex are the open sets $U_{i}$ .
A finite set of vertices $U_{i} 1, U_{i} 2, . . ., U_{i} k$ form a k-simplex if and only if their intersection $U_{i} 1 \cap U_{i} 2 \cap . . . \cap U_{i} k$ is non-empty.

In other words, the Čech complex consists of all possible intersections of the open sets in the cover, including the empty intersection. The complex captures the topology of the underlying space, and its homology groups can be used to compute topological invariants such as the Betti numbers.

The Čech complex is named after the Czech mathematician Eduard Čech, who introduced it in the 1920s as a tool for studying the topology of metric spaces. Since then, the Čech complex has been generalised to other settings, such as sheaf theory and algebraic geometry, and has found applications in various areas of mathematics and computer science.