Topology

• A set $A\subseteq {\mathbb{R}}^2$ is called closed if for every convergent sequence $(p_n)$ of points in $A$, we have $\underset{n\to \infty }{\lim }{p}_n\in A$.
• A point $p\in A\subseteq {\mathbb{R}}^2$ is called an interior point of $A$ if there exists a neighborhood $N(p)$ such that $N(p)\subseteq A$.
• A point $p\in A\subseteq {\mathbb{R}}^2$ is called a boundary point of a closed set $A$ if $p$ is not an interior point of $A$.
• The set of all interior points of $A$ is called the interior of $A$ and is denoted by $\text{int}(A)$.
• The set of all boundary points of a closed set $A$ is called the boundary of $A$ and is denoted by $\partial A$.
• A set $A\subseteq {\mathbb{R}}^2$ is called open if all its points are interior points.
• A set $A\subseteq {\mathbb{R}}^2$ is open if and only if $A^{\complement }={\mathbb{R}}^2\setminus A$ is closed.
• Let $p\in {\mathbb{R}}^2$ and $r>0$. Every point $q\in N_r(p)$ is an interior point of $N_r(p)$.

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• A metric space is a pair $(X,d)$, where $X$ is a set and $d:X\times X\to \mathbb{R}$ is a distance function satisfying:

  • Non-negativity: $d(x,y)\ge 0$ $\forall x,y\in X$,
  • Identity of indiscernibles: $d(x,y)=0$ iff $x=y$,
  • Symmetry: $d(x,y)=d(y,x)$ $\forall x,y\in X$,
  • Triangle inequality: $d(x,z)\le d(x,y)+d(y,z)$ $\forall x,y,z\in X$.

• In a metric space $(X,d)$, a set $A$ is open if $\forall x\in A,\exists \epsilon >0:B_{\epsilon }(x)\subset A$. $A$ is closed if $X\setminus A$ is open.

• $a$ is said to be a limit point (or accumulation point) of a set $A$ if $\forall \epsilon >0,\exists x\ne a\in V_{\epsilon }(a)\cap A$

• isolated point definitions, $x\in S$ is a point in a topological space $X$

  • A point $x$ is an isolated point of $S$
  • There exists a neighborhood of $x$ that does not contain any other points of $S$
  • $\{x\}$ is an open set in the topological space $S$ (considered as a subspace of $X$).
  • $x\in S$ is not a limit point of $S$

• A function $f:X\to Y$ is continuous at a point $x$ if $\forall$ open set $V$ containing $f(x)$, $\exists$ open set $U$ containing $x$ s.t. $f(U)\subset V$.

• A subset $K$ of a metric space $(X,d)$ is compact if every open cover of $K$ has a finite subcover.

  • Heine–Borel theorem - For any subset $A$ of Euclidean space, $A$ is compact if and only if it is closed and bounded; this is the