Christina DanielThis is a summary of Chapter 1, Section 1 of Reference [1], which is Equations of Mathematical Physics by V.S. Vladimirov. The majority of this post consists of definitions from V.S. Vladimirov.
Terms in Set Theory
Foundation
$A$ is a set.
$a \in A$ means element $a$ is contained in set $A$.
$B$ is another set.
$A \subset B$ means $A$ is contained in $B$.
$A \cup B$ means the union of $A$ and $B$.
$A \cap B$ means the intersection of $A$ and $B$.
$A \backslash B$ means the complement of $B$ relative to $A$.
$A \times B$ is called the product of $A$ and $B$ and means the set of pairs $(a,b)$, $a \in A$, $b \in B$.
$\varnothing$ is the symbol for an empty set.
$A=B$ means $A$ is coincident with $B$.
Points
$R^n$ means $n$-dimensional Euclidean space.
$x = (x_1,x_2,…,x_n)$ is the set of points belonging to the set $R^n$.
- Perhaps this $x$ should have an index such as $\gamma$, to keep track of all the different points.
$(x,y) = x_1 y_1 + x_2 y_2 + … + x_n y_n$ is called a scalar product.
$|x| = \sqrt{x_1^2 + x_2^2 + … + x_n^2}$ is called the norm of $x$.
“$|x-y|$ is the Euclidean distance between $x$ and $y$.”
“$|x – x_0| < R$ is called an open sphere with radius $R$ and center $x_0$ and denoted by $U(x_0; R)$.” Also, $U_R = U(0;R)$.
- Note that if $R=0$, $|x – x_0| < 0$, but $|x – x_0| \ge 0$ by definition, so $R$ cannot equal zero. $R$ must be positive to satisfy the inequality; $R>0$
Sequence of Points
“The sequence of points $x_k = (x_{1k}, x_{2k}, x_{3k}, …, x_{nk})$, $k=1,2,…,$ is said to converge to the point $x$ in $R^n$, written $x_k \rightarrow x$ as $k \rightarrow \infty$, if $|x_k – x| \rightarrow 0$ as $k \rightarrow \infty$.”
- Note that $k$ is an index. It helps to picture $k$ as a independent variable on a line, with $x_k$ as a dependent variable.
“The sequence $x_k, k=1,2,…,$ is said to converge in itself in $R^n$ if $|x_k – x_p| \rightarrow 0$ as $k \rightarrow \infty$, $p \rightarrow \infty$.”
Cauchy’s Principle of Convergence
“Cauchy’s Principle of convergence: In order that a sequence of points should converge in $R^n$, it is necessary and sufficient that it should converge in itself in $R^n$.”
Necessary and Sufficient Conditions
Reference [2] provides definitions of necessary and sufficient conditions. They are as follows.
“If A, then B” is called a necessary condition.
“A, only if B” is called a sufficient condition.
Bounded, Open, Connected, Closed, Compactum
“A set is said to be bounded in $R^n$ if there is a sphere containing this set.”
“$x_0$ is called an interior point of a set if there is a sphere $U(x_0;R)$ contained in this set.”
“A set is called open if all its points are interior.”
“A set is called connected if any two of its points can be joined by an unbroken line lying this set.”
- I assume this line can be curved.
“A connected open set is called a region.”
“The point $x_0$ is called a limit point of the set $A$ if there is a sequence $x_k, k=1,2,…,$ such that $x_k \in A$, $x_k \rightarrow x_0$ as $k \rightarrow \infty$.”
“If we add all limit points of a set $A$ to the set $A$, the set obtained is called the closure of the set A and is denoted by $\bar{A}$; it is clear that $A \subset \bar{A}$.”
“If a set coincides with its closure, it is called closed.”
“A closed bounded set is called a compactum.”
Neighborhood of a Set
“Each open set containing $A$ is called a neighborhood of the set $A$.”
“The union of the spheres $U(x; \varepsilon)$, when $x$ ranges over $A$, written symbolically as $A_{\in} = \cup_{x \in A} U(x;\varepsilon)$, is called an $\varepsilon$-neighborhood of the set $A$.”
Characteristic Function of a Set
“The function $\chi_{A}(x)$, which is equal to 1 when $x \in A$ and to 0 when $x \not\in A$, is called the characteristic function of the set $A$.”
Comment
Regardless of how close two different points are in physical space, these points have coordinates which are numerically different, so there is a nonzero Euclidean distance between these two points. This nonzero distance can be thought of as the radius of an open sphere.
Boundary of a Region
“Let $G$ be a region.”
- Recall that a region is connected and open.
- A set is open if all points are interior–that is, an open sphere can be centered around each point.
“The points of”
“closure $\bar{G}$ not belonging to $G$”
“form a closed set $S$, called the boundary of the region $G$, so that
$S = \bar{G} \backslash G$.”
“The connected part of the set $S \cap U(x_0;R)$ which contains the point $x_0$ is called the neighborhood of the point $x_0$ on the surface $S$.
- Do not confuse a neighborhood of a set with the neighborhood of a point on a surface.