Let V = \{v_0, \cdots, v_n\} be a set of points in \mathbb R^m. We say V is in general position if it is not contained in an affine subspace of dimension n-1 in \mathbb R^n.
When V is in general position, we define the simplex spanned by V as the convex hull of V:
r_0v_0 + r_1v_1 + \cdots + r_nv_n
where \sum_{i=0}^n r_i = 1 and r_i \ge 0 for all i.
The points of V are called vertices and the dimension of the simplex is n.
If W is a subset of V then a simplex spanned by W is a face of the simplex spanned by V.
A simplicial complex is a space X which is a union of a list L of simplices with the following two properties:
A subcomplex of a simplicial complex X is a collection of simplices belonging to X that is a simplicial complex in its own right.
A pair (V, \Sigma) where V is a finite set called the vertex set and where \Sigma is a family of nonempty subsets of V such that
\sigma \in \Sigma \land \emptyset \ne \tau \subseteq \sigma \implies \tau \in \Sigma
The elements of \Sigma are called simplices.
For a simplicial complex X, the associated abstract simplicial complex has the same vertex set V, and a set of vertices is in \Sigma iff it spans a simplex in X.
A chain complex V_* = (V_*, \partial_*) is a sequence \cdots, V_i, \cdots, V_0 of vector spaces together with a sequence of linear transformations \partial_i:V_i \to V_{i-1} such that \partial_i \circ \partial_{i+1} = 0. The elements of a given V_i are called i-chains, and the maps \partial_i are called boundary maps or differentials.
Let \mathbb k be a field and let X be a set. Let F(X) be a vector space on the set X, i.e. a vector space over \mathbb k whose basis is in bijective correspondence with X.
Given an abstract simplicial complex \Sigma and a field \mathbb k we define a simplicial chain complex C_*(\Sigma) by letting C_n(\Sigma) be the vector space on the set of n-dimensional faces of \Sigma. The boundary maps are:
\partial_n((v_0, \cdots, v_n)) = \sum_{i=0}^n (-1)^i (v_0, \cdots, \hat v_i, \cdots, v_n)
where \hat v_i denotes omitting that term (for example (v_0, v_1, \hat v_2, v_3) = (v_0, v_1, v_3)).
Let Z be a finite subset of \mathbb R^n (point cloud). The Vietoris-Rips complex at scale \varepsilon > 0 is a simplicial complex with vertex set Z for which a family \{z_0, \cdots, z_k\} spans a simplex if d(z_i, z_j) \le \varepsilon for 0 \le i < j \le k. Denoted by \text{VR}(Z, \varepsilon).
See here for a definition of a covering and a nerve.
Given a point cloud Z \subset \mathbb R^n at scale \epsilon we define the covering U_\epsilon^{\text{Cech}} to be \{B(z, \epsilon)\}_{z \in Z}. The Čech complex is the nerve N(U_\epsilon^{\text{Cech}}), denoted by C^{\text{Cech}}(Z, \epsilon).
Given a finite subset X \subset \mathbb R^n the Voronoi cell of x \in X is the set V(x) = \{p \in \mathbb R^n : \forall_{z \in X} d(p, x) \le d(p, z)\}. \{V(x)\}_{x \in X} forms a covering of \mathbb R^n called the Voronoi covering denoted by U_X^{\text{Vor}}. The nerve N(U_X^{\text{Vor}}) is the Delaunay complex.