# Theory Glossary¶

## Symbols¶

 $$\Bbbk$$ An arbitrary field. $$\mathbb R$$ The field of real numbers. $$\overline{\mathbb R}$$ The two point compactification $$[-\infty, +\infty]$$ of the real numbers. $$\mathbb N$$ The counting numbers $$0,1,2, \ldots$$ as a subset of $$\mathbb R$$. $$\mathbb R^d$$ The vector space of $$d$$-tuples of real numbers. $$\Delta$$ The multiset $$\lbrace ( s, s) \mid s \in \mathbb{R} \rbrace$$ with multiplicity $$( s,s ) \mapsto +\infty$$.

## Analysis¶

### Metric space¶

A set $$X$$ with a function

$d : X \times X \to \mathbb R$

is said to be a metric space if the values of $$d$$ are all non-negative and for all $$x,y,z \in X$$

$d(x,y) = 0\ \Leftrightarrow\ x = y$
$d(x,y) = d(y,x)$
$d(x,z) \leq d(x,y) + d(y, z).$

In this case the $$d$$ is referred to as the metric or the distance function.

### Normed space¶

A vector space $$V$$ together with a function

$||-|| : V \to \mathbb R$

is said to be an normed space if the values of $$||-||$$ are all non-negative and for all $$u,v \in V$$ and $$a \in \mathbb R$$

$||u|| = 0\ \Leftrightarrow\ u = 0$
$||a u || = |a|\, ||u||$
$||u + v|| \leq ||u|| + ||v||.$

The function $$||-||$$ is referred to as the norm.

A normed space is naturally a metric space with distance function

$d(u,v) = ||u-v||.$

### Inner product space¶

A vector space $$V$$ together with a function

$\langle -, - \rangle : V \times V \to \mathbb R$

is said to be an inner product space if for all $$u,v,w \in V$$ and $$a \in \mathbb R$$

$u \neq 0\ \Rightarrow\ \langle u, u \rangle > 0$
$\langle u, v\rangle = \langle v, u\rangle$
$\langle au+v, w \rangle = a\langle u, w \rangle + \langle v, w \rangle.$

The function $$\langle -, - \rangle$$ is referred to as the inner product.

An inner product space is naturally a normed space with

$||u|| = \sqrt{\langle u, u \rangle}.$

### Vectorization, amplitude and kernel¶

Let $$X$$ be a set, for example, the set of all persistence diagrams. A vectorization for $$X$$ is a function

$\phi : X \to V$

where $$V$$ is a vector space.

An amplitude on $$X$$ is a function

$A : X \to \mathbb R$

for which there exists a vectorization $$\phi : X \to V$$ with $$V$$ a normed space such that

$A(x) = ||\phi(x)||$

for all $$x \in X$$.

A kernel on the set $$X$$ is a function

$k : X \times X \to \mathbb R$

for which there exists a vectorization $$\phi : X \to V$$ with $$V$$ an inner product space such that

$k(x,y) = \langle \phi(x), \phi(y) \rangle$

for each $$x,y \in X$$.

### Euclidean distance and $$l^p$$-norms¶

The vector space $$\mathbb R^n$$ is an inner product space with inner product

$\langle x, y \rangle = (x_1-y_1)^2 + \cdots + (x_n-y_n)^2.$

This inner product is referred to as dot product and the associated norm and distance function are respectively named Euclidean norm and Euclidean distance.

For any $$p \in (0,\infty]$$ the pair $$\mathbb R^n, ||-||_p$$ with

$||x||_p = (x_1^p + \cdots + x_n^p)^{1/p}$

if $$p$$ is finite and

$||x||_{\infty} = max\{x_i\ |\ i = 1,\dots,n\}$

is a normed spaced and its norm is referred to as the $$l^p$$-norm.

### Distance matrices and point clouds¶

Let $$(X, d)$$ be a finite metric space. A distance matrix associated to it is obtained by choosing a total order on $$X = {x_1 < \cdots < x_m}$$ and setting the $$(i,j)$$-entry to be equal to $$d(x_i, x_j)$$.

A point cloud is a finite subset of $$\mathbb{R}^n$$ (for some $$n$$) together with the metric induced from the Euclidean distance.

### $$L^p$$-norms¶

Let $$U \subseteq \mathbb R^n$$ and $$C(U, \mathbb R)$$ be the set of continuous real-valued functions on $$U$$. A function $$f \in C(U, \mathbb R)$$ is said to be $$p$$-integrable if

$\int_U |f(x)|^p dx$

is finite. The subset of $$p$$-integrable functions together with the assignment $$||-||_p$$

$f \mapsto \left( \int_U |f(x)|^p dx \right)^{1/p}$

is a normed space and $$||-||_p$$ is referred to as the $$L^p$$-norm.

The only $$L^p$$-norm that is induced from an inner product is $$L^2$$, and the inner product is given by

$\langle f, g \rangle = \left(\int_U |f(x)-g(x)|^2 dx\right)^{1/2}.$

## Homology¶

### Cubical complex¶

An elementary interval $$I_a$$ is a subset of $$\mathbb{R}$$ of the form $$[a, a+1]$$ or $$[a,a] = \{a\}$$ for some $$a \in \mathbb{R}$$. These two types are called respectively non-degenerate and degenerate. To a non-degenerate elementary interval we assign two degenerate elementary intervals

$d^+ I_a = \lbrack a+1, a+1 \rbrack \qquad \text{and} \qquad d^- I_a = \lbrack a, a \rbrack.$

An elementary cube is a subset of the form

$I_{a_1} \times \cdots \times I_{a_N} \subset \mathbb{R}^N$

where each $$I_{a_i}$$ is an elementary interval. We refer to the total number of its non-degenerate factors $$I_{a_{k_1}}, \dots, I_{a_{k_n}}$$ as its dimension and, assuming

$a_{k_1} < \cdots < a_{k_{n,}}$

we define for $$i = 1, \dots, n$$ the following two elementary cubes

$d_i^\pm I^N = I_{a_1} \times \cdots \times d^\pm I_{a_{k_i}} \times \cdots \times I_{a_{N}}.$

A cubical complex is a finite set of elementary cubes of $$\mathbb{R}^N$$, and a subcomplex of $$X$$ is a cubical complex whose elementary cubes are also in $$X$$.

#### Reference:¶

(Kaczynski, Mischaikow, and Mrozek 2004)

### Simplicial complex¶

A set $$\{v_0, \dots, v_n\} \subset \mathbb{R}^N$$ is said to be geometrically independent if the vectors $$\{v_0-v_1, \dots, v_0-v_n\}$$ are linearly independent. In this case, we refer to their convex closure as a simplex, explicitly

$\lbrack v_0, \dots , v_n \rbrack = \left\{ \sum c_i (v_0 - v_i)\ \big|\ c_1+\dots+c_n = 1,\ c_i \geq 0 \right\}$

and to $$n$$ as its dimension. The $$i$$-th face of $$\lbrack v_0, \dots, v_n \rbrack$$ is defined by

$d_i[v_0, \ldots, v_n] = [v_0, \dots, \widehat{v}_i, \dots, v_n]$

where $$\widehat{v}_i$$ denotes the absence of $$v_i$$ from the set.

A simplicial complex $$X$$ is a finite union of simplices in $$\mathbb{R}^N$$ satisfying that every face of a simplex in $$X$$ is in $$X$$ and that the non-empty intersection of two simplices in $$X$$ is a face of each. Every simplicial complex defines an abstract simplicial complex.

### Abstract simplicial complex¶

An abstract simplicial complex is a pair of sets $$(V, X)$$ with the elements of $$X$$ being subsets of $$V$$ such that:

1. for every $$v$$ in $$V$$, the singleton $$\{v\}$$ is in $$X$$ and

2. if $$x$$ is in $$X$$ and $$y$$ is a subset of $$x$$, then $$y$$ is in $$X$$.

We abuse notation and denote the pair $$(V, X)$$ simply by $$X$$.

The elements of $$X$$ are called simplices and the dimension of a simplex $$x$$ is defined by $$|x| = \# x - 1$$ where $$\# x$$ denotes the cardinality of $$x$$. Simplices of dimension $$d$$ are called $$d$$-simplices. We abuse terminology and refer to the elements of $$V$$ and to their associated $$0$$-simplices both as vertices.

A simplicial map between simplicial complexes is a function between their vertices such that the image of any simplex via the induced map is a simplex.

A simplicial complex $$X$$ is a subcomplex of a simplicial complex $$Y$$ if every simplex of $$X$$ is a simplex of $$Y$$.

Given a finite abstract simplicial complex $$X = (V, X)$$ we can choose a bijection from $$V$$ to a geometrically independent subset of $$\mathbb R^N$$ and associate a simplicial complex to $$X$$ called its geometric realization.

### Ordered simplicial complex¶

An ordered simplicial complex is an abstract simplicial complex where the set of vertices is equipped with a partial order such that the restriction of this partial order to any simplex is a total order. We denote an $$n$$-simplex using its ordered vertices by $$\lbrack v_0, \dots, v_n \rbrack$$.

A simplicial map between ordered simplicial complexes is a simplicial map $$f$$ between their underlying simplicial complexes preserving the order, i.e., $$v \leq w$$ implies $$f(v) \leq f(w)$$.

### Directed simplicial complex¶

A directed simplicial complex is a pair of sets $$(V, X)$$ with the elements of $$X$$ being tuples of elements of $$V$$, i.e., elements in $$\bigcup_{n\geq1} V^{\times n}$$ such that:

1. for every $$v$$ in $$V$$, the tuple $$v$$ is in $$X$$

2. if $$x$$ is in $$X$$ and $$y$$ is a subtuple of $$x$$, then $$y$$ is in $$X$$.

With appropriate modifications the same terminology and notation introduced for ordered simplicial complex applies to directed simplicial complex.

### Chain complex¶

A chain complex of is a pair $$(C_*, \partial)$$ where

$C_* = \bigoplus_{n \in \mathbb Z} C_n \quad \mathrm{and} \quad \partial = \bigoplus_{n \in \mathbb Z} \partial_n$

with $$C_n$$ a $$\Bbbk$$-vector space and $$\partial_n : C_{n+1} \to C_n$$ is a $$\Bbbk$$-linear map such that $$\partial_{n+1} \partial_n = 0$$. We refer to $$\partial$$ as the boundary map of the chain complex.

The elements of $$C$$ are called chains and if $$c \in C_n$$ we say its degree is $$n$$ or simply that it is an $$n$$-chain. Elements in the kernel of $$\partial$$ are called cycles, and elements in the image of $$\partial$$ are called boundaries. Notice that every boundary is a cycle. This fact is central to the definition of homology.

A chain map is a $$\Bbbk$$-linear map $$f : C \to C'$$ between chain complexes such that $$f(C_n) \subseteq C'_n$$ and $$\partial f = f \partial$$.

Given a chain complex $$(C_*, \partial)$$, its linear dual $$C^*$$ is also a chain complex with $$C^{-n} = \mathrm{Hom_\Bbbk}(C_n, \Bbbk)$$ and boundary map $$\delta$$ defined by $$\delta(\alpha)(c) = \alpha(\partial c)$$ for any $$\alpha \in C^*$$ and $$c \in C_*$$.

### Homology and cohomology¶

Let $$(C_*, \partial)$$ be a chain complex. Its $$n$$-th homology group is the quotient of the subspace of $$n$$-cycles by the subspace of $$n$$-boundaries, that is, $$H_n(C_*) = \mathrm{ker}(\partial_n)/ \mathrm{im}(\partial_{n+1})$$. The homology of $$(C, \partial)$$ is defined by $$H_*(C) = \bigoplus_{n \in \mathbb Z} H_n(C)$$.

When the chain complex under consideration is the linear dual of a chain complex we sometimes refer to its homology as the cohomology of the predual complex and write $$H^n$$ for $$H_{-n}$$.

A chain map $$f : C \to C'$$ induces a map between the associated homologies.

### Simplicial chains and simplicial homology¶

Let $$X$$ be an ordered or directed simplicial complex and denote the subset of $$n$$-simplices by $$X_n$$. Define its simplicial chain complex with $$\Bbbk$$-coefficients $$C_*(X; \Bbbk)$$ by

$C_n(X; \Bbbk) = \Bbbk\{X_n\}, \qquad \partial_n(x) = \sum_{i=0}^{n} (-1)^i d_ix$

and its homology and cohomology with $$\Bbbk$$-coefficients as the homology and cohomology of this chain complex. We use the notation $$H_*(X; \Bbbk)$$ and $$H^*(X; \Bbbk)$$ for these.

A simplicial map induces a chain map between the associated simplicial chain complexes and, therefore, between the associated simplicial (co)homologies.

### Cubical chains and cubical homology¶

Let $$X$$ be a cubical complex and denote the subset of $$n$$-cubes by $$X_n$$. Define the cubical chain complex with $$\Bbbk$$-coefficients $$C_*(X; \Bbbk)$$ by

$C_n(X; \Bbbk) = \Bbbk\{X_n\}, \qquad \partial_n x = \sum_{i = 1}^{n} (-1)^{i-1}(d^+_i x - d^-_i x)$

where $$x = I_1 \times \cdots \times I_N$$ and $$s(i)$$ is the dimension of $$I_1 \times \cdots \times I_i$$. Its homology and cohomology with $$\Bbbk$$-coefficients is the homology and cohomology of this chain complex. We use the notation $$H_*(X; \Bbbk)$$ and $$H^*(X; \Bbbk)$$ for these.

## Persistence¶

### Filtered complex¶

A filtered complex is a collection of simplicial or cubical complexes $$\{X_s\}_{s \in \mathbb R}$$ such that $$X_s$$ is a subcomplex of $$X_t$$ for each $$s \leq t$$.

### Cellwise filtration¶

A cellwise filtration is a simplicial or cubical complex $$X$$ together with a total order $$\leq$$ on its simplices or elementary cubes such that for each $$y \in X$$ the set $$\{x \in X\ :\ x \leq y\}$$ is a subcomplex of $$X$$. A cellwise filtration can be naturally thought of as a filtered complex.

### Clique and flag complexes¶

Let $$G$$ be a $$1$$-dimensional abstract (resp. directed) simplicial complex. The abstract (resp. directed) simplicial complex $$\langle G \rangle$$ has the same set of vertices as $$G$$ and $$\{v_0, \dots, v_n\}$$ (resp. $$(v_0, \dots, v_n)$$) is a simplex in $$\langle G \rangle$$ if an only if $$\{v_i, v_j\}$$ (resp. $$(v_i, v_j)$$) is in $$G$$ for each pair of vertices $$v_i, v_j$$.

An abstract (resp. directed) simplicial complex $$X$$ is a clique (resp. flag) complex if $$X = \langle G \rangle$$ for some $$G$$.

Given a function

$w : G \to \mathbb R \cup \{\infty\}$

consider the extension

$w : \langle G \rangle \to \mathbb R \cup \{\infty\}$

defined respectively by

\begin{split}\begin{aligned} w\{v_0, \dots, v_n\} & = \max\{ w\{v_i, v_j\}\ |\ i \neq j\} \\ w(v_0, \dots, v_n) & = \max\{ w(v_i, v_j)\ |\ i < j\} \end{aligned}\end{split}

and define the filtered complex $$\{\langle G \rangle_{s}\}_{s \in \mathbb R}$$ by

$\langle G \rangle_s = \{\sigma \in \langle G \rangle\ |\ w(\sigma) \leq s\}.$

A filtered complex $$\{X_s\}_{s \in \mathbb R}$$ is a filtered clique (resp. flag) complex if $$X_s = \langle G \rangle_s$$ for some $$(G,w)$$.

### Persistence module¶

A persistence module is a collection containing a $$\Bbbk$$-vector spaces $$V(s)$$ for each real number $$s$$ together with $$\Bbbk$$-linear maps $$f_{st} : V(s) \to V(t)$$, referred to as structure maps, for each pair $$s \leq t$$, satisfying naturality, i.e., if $$r \leq s \leq t$$, then $$f_{rt} = f_{st} \circ f_{rs}$$ and tameness, i.e., all but finitely many structure maps are isomorphisms.

A morphism of persistence modules $$F : V \to W$$ is a collection of linear maps $$F(s) : V(s) \to W(s)$$ such that $$F(t) \circ f_{st} = f_{st} \circ F(s)$$ for each par of reals $$s \leq t$$. We say that $$F$$ is an isomorphisms if each $$F(s)$$ is.

### Persistent simplicial (co)homology¶

Let $$\{X(s)\}_{s \in \mathbb R}$$ be a set of ordered or directed simplicial complexes together with simplicial maps $$f_{st} : X(s) \to X(t)$$ for each pair $$s \leq t$$, such that

$r \leq s \leq t\ \quad\text{implies} \quad f_{rt} = f_{st} \circ f_{rs}$

for example, a filtered complex. Its persistent simplicial homology with $$\Bbbk$$-coefficients is the persistence module

$H_*(X(s); \Bbbk)$

with structure maps $$H_*(f_{st}) : H_*(X(s); \Bbbk) \to H_*(X(t); \Bbbk)$$ induced form the maps $$f_{st}.$$ In general, the collection constructed this way needs not satisfy the tameness condition of a persistence module, but we restrict attention to the cases where it does. Its persistence simplicial cohomology with $$\Bbbk$$-coefficients is defined analogously.

### Vietoris-Rips complex and Vietoris-Rips persistence¶

Let $$(X, d)$$ be a finite metric space. Define the Vietoris-Rips complex of $$X$$ as the filtered complex $$VR_s(X)$$ that contains a subset of $$X$$ as a simplex if all pairwise distances in the subset are less than or equal to $$s$$, explicitly

$VR_s(X) = \Big\{ \lbrack v_0,\dots,v_n \rbrack \ \Big|\ \forall i,j\ \,d(v_i, v_j) \leq s \Big\}.$

The Vietoris-Rips persistence of $$(X, d)$$ is the persistent simplicial (co)homology of $$VR_s(X)$$.

A more general version is obtained by replacing the distance function with an arbitrary function

$w : X \times X \to \mathbb R \cup \{\infty\}$

and defining $$VR_s(X)$$ as the filtered clique complex associated to $$(X \times X ,w)$$.

### Čech complex and Čech persistence¶

Let $$(X, d)$$ be a point cloud. Define the Čech complex of $$X$$ as the filtered complex $$\check{C}_s(X)$$ that is empty if $$s<0$$ and, if $$s \geq 0$$, contains a subset of $$X$$ as a simplex if the balls of radius $$s$$ with centers in the subset have a non-empty intersection, explicitly

$\check{C}_s(X) = \Big\{ \lbrack v_0,\dots,v_n \rbrack \ \Big|\ \bigcap_{i=0}^n B_s(x_i) \neq \emptyset \Big\}.$

The Čech persistence (co)homology of $$(X,d)$$ is the persistent simplicial (co)homology of $$\check{C}_s(X)$$.

### Multiset¶

A multiset is a pair $$(S, \phi)$$ where $$S$$ is a set and $$\phi : S \to \mathbb N \cup \{+\infty\}$$ is a function attaining positive values. For $$s \in S$$ we refer to $$\phi(s)$$ as its multiplicity. The union of two multisets $$(S_1, \phi_1), (S_2, \phi_2)$$ is the multiset $$(S_1 \cup S_2, \phi_1 \cup \phi_2)$$ with

$\begin{split}(\phi_1 \cup \phi_2)(s) = \begin{cases} \phi_1(s) & s \in S_1, s \not\in S_2 \\ \phi_2(s) & s \in S_2, s \not\in S_1 \\ \phi_1(s) + \phi_2(s) & s \in S_1, s \in S_2. \\ \end{cases}\end{split}$

### Persistence diagram¶

A persistence diagram is a multiset of points in

$\mathbb R \times \big( \mathbb{R} \cup \{+\infty\} \big).$

Given a persistence module, its associated persistence diagram is determined by the following condition: for each pair $$s,t$$ the number counted with multiplicity of points $$(b,d)$$ in the multiset, satisfying $$b \leq s \leq t < d$$ is equal to the rank of $$f_{st}.$$

A well known result establishes that there exists an isomorphism between two persistence module if and only if their persistence diagrams are equal.

### Wasserstein and bottleneck distance¶

The $$p$$-Wasserstein distance between two persistence diagrams $$D_1$$ and $$D_2$$ is the infimum over all bijections $$\gamma: D_1 \cup \Delta \to D_2 \cup \Delta$$ of

$\Big(\sum_{x \in D_1 \cup \Delta} ||x - \gamma(x)||_\infty^p \Big)^{1/p}$

where $$||-||_\infty$$ is defined for $$(x,y) \in \mathbb R^2$$ by $$\max\{|x|, |y|\}$$.

The limit $$p \to \infty$$ defines the bottleneck distance. More explicitly, it is the infimum over the same set of bijections of the value

$\sup_{x \in D_1 \cup \Delta} ||x - \gamma(x)||_{\infty}.$

The set of persistence diagrams together with any of the distances above is a metric space.

#### Reference:¶

(Kerber, Morozov, and Nigmetov 2017)

### Persistence landscape¶

Let $$\{(b_i, d_i)\}_{i \in I}$$ be a persistence diagram. Its persistence landscape is the set $$\{\lambda_k\}_{k \in \mathbb N}$$ of functions

$\lambda_k : \mathbb R \to \overline{\mathbb R}$

defined by letting $$\lambda_k(t)$$ be the $$k$$-th largest value of the set $$\{\Lambda_i(t)\}_ {i \in I}$$ where

$\Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+$

and $$c_+ := \max(c,0)$$. The function $$\lambda_k$$ is referred to as the $$k$$-layer of the persistence landscape.

We describe the graph of each $$\lambda_k$$ intuitively. For each $$i \in I$$, draw an isosceles triangle with base the interval $$(b_i, d_i)$$ on the horizontal $$t$$-axis, and sides with slope 1 and $$-1$$. This subdivides the plane into a number of polygonal regions. Label each of these regions by the number of triangles containing it. If $$P_k$$ is the union of the polygonal regions with values at least $$k$$, then the graph of $$\lambda_k$$ is the upper contour of $$P_k$$, with $$\lambda_k(a) = 0$$ if the vertical line $$t=a$$ does not intersect $$P_k$$.

The persistence landscape construction defines a vectorization of the set of persistence diagrams with target the vector space of real-valued function on $$\mathbb N \times \mathbb R$$. For any $$p = 1,\dots,\infty$$ we can restrict attention to persistence diagrams $$D$$ whose associated persistence landscape $$\lambda$$ is $$p$$-integrable, that is to say,

$\label{equation:persistence_landscape_norm} ||\lambda||_p = \left( \sum_{i \in \mathbb N} ||\lambda_i||^p_p \right)^{1/p}$

where

$||\lambda_i||_p = \left( \int_{\mathbb R} \lambda_i^p(x)\, dx \right)^{1/p}$

is finite. In this case we refer to [equation:persistence_landscape_norm] as the $$p$$-landscape norm of $$D$$ and, for $$p = 2$$, define the value of the landscape kernel on two persistence diagrams $$D$$ and $$E$$ as

$\langle \lambda, \mu \rangle = \left(\sum_{i \in \mathbb N} \int_{\mathbb R} |\lambda_i(x) - \mu_i(x)|^2\, dx\right)^{1/2}$

where $$\lambda$$ and $$\mu$$ are their associated persistence landscapes.

## Time series¶

### Time series¶

A time series is a sequence $$\{y_i\}_{i = 0}^n$$ of real numbers.

A common construction of a times series $$\{x_i\}_{i = 0}^n$$ is given by choosing $$x_0$$ arbitrarily as well as a step parameter $$h$$ and setting

$x_i = x_0 + h\cdot i.$

Another usual construction is as follows: given a time series $$\{x_i\}_{i = 0}^n \subseteq U$$ and a function

$f : U \subseteq \mathbb R \to \mathbb R$

we obtain a new time series $$\{f(x_i)\}_{i = 0}^n$$.

Generalizing the previous construction we can define a time series from a function

$\varphi : U \times M \to M, \qquad U \subseteq \mathbb R, \qquad M \subseteq \mathbb R^d$

using a function $$f : M \to \mathbb R$$ as follows: let $$\{t_i\}_{i=0}^n$$ be a time series taking values in $$U$$, then

$\{f(\varphi(t_i, m))\}_{i=0}^n$

for an arbitrarily chosen $$m \in M$$.

### Takens embedding¶

Let $$M \subset \mathbb R^d$$ be a compact manifold of dimension $$n$$. Let

$\varphi : \mathbb R \times M \to M$

and

$f : M \to \mathbb R$

be generic smooth functions. Then, for any $$\tau > 0$$ the map

$M \to \mathbb R^{2n+1}$

defined by

$x \mapsto\big( f(x), f(x_1), f(x_2), \dots, f(x_{2n}) \big)$

where

$x_i = \varphi(i \cdot \tau, x)$

is an injective map with full rank.

(Takens 1981)

### Manifold¶

Intuitively, a manifold of dimension $$n$$ is a space locally equivalent to $$\mathbb R^n$$. Formally, a subset $$M$$ of $$\mathbb R^d$$ is an $$n$$-dimensional manifold if for each $$x \in M$$ there exists an open ball $$B(x) = \{ y \in M\,;\ d(x,y) < \epsilon\}$$ and a smooth function with smooth inverse

$\phi_x : B(x) \to \{v \in \mathbb R^n\,;\ ||v||<1\}.$

#### References:¶

(Milnor and Weaver 1997; Guillemin and Pollack 2010)

### Compact subset¶

A subset $$K$$ of a metric space $$(X,d)$$ is said to be bounded if there exist a real number $$D$$ such that for each pair of elements in $$K$$ the distance between them is less than $$D$$. It is said to be complete if for any $$x \in X$$ it is the case that $$x \in K$$ if for any $$\epsilon > 0$$ the intersection between $$K$$ and $$\{y \,;\ d(x,y) < \epsilon \}$$ is not empty. It is said to be compact if it is both bounded and complete.

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