Open and Closed Sets in R

Subjects: Real Analysis
Links: Real Numbers

Open Sets

Def: We define an $\epsilon$ -neighborhood of $a$ is the set

Def: A set $U\subseteq \mathbb{R}$ is called open if for all points $a\in U$ there exist an $\epsilon$-neighborhood $B_{\epsilon }(a)\subseteq O$

Th: The union of an arbitrary collection of open sets is open.

The intersection of a finite collection of open sets is open

$\mathbb{R}$ and $\varnothing$ are open sets

Thus we just defined a topology on $\mathbb{R}$ through the norm $|\cdot |$.

Every system of mutually disjoint open intervals in $\mathbb{R}$ is at most countable

Every nonempty open set of real numbers has cardinality $2^{{\mathrm{\aleph }}_0}$

There are $2^{{\mathrm{\aleph }}_0}$ open sets

Closed sets

Def: A point $x$ is a limit point of a set $A$, if for every $\epsilon$ -neighborhood of $x$ intersects the set $A$ at some other point other than $x$. This can be written with mathematical symbols as

Th: A point $x$ is a limit point of a set $A$ iff $x=\underset{n\to \mathrm{\infty }}{lim}{a}_n$ for some sequence $(a_n{)}_{n\in \mathbb{N}}\subseteq A$ satisfying $a_n\ne x$ for all $n\in \mathbb{N}$.

Def: A point $a\in A$ is an isolated point of $A$ if is not a limit point of $A$. The set of all limit points of $A$ is denoted as $A^{\prime }$. The set of isolated points of $A$ is $A\setminus A^{\prime }$.

Def: A set $F\subseteq \mathbb{R}$ is closed if it contains its limit points, meaning $F^{\prime }\subseteq F$ . This notion in $\mathbb{R}$ is equivalent to the topological definition of closed sets.

Th: A set $F\subseteq \mathbb{R}$ is closed iff for every Cauchy sequence contained in $F$ has a limit that is also contained in $F$.

******Th (Density of $\mathbb{Q}$ in $\mathbb{R}$): For every $y\in \mathbb{R}$, there exists a sequence of rational numbers that converges to $y$.

Def: Given $A\subseteq \mathbb{R}$, the closure of $A$ is defined as $\mathrm{cl}(A)=\bar{A}=A\cup A^{\prime }$

Th: For any $A\subseteq \mathbb{R}$, the closure $\mathrm{cl}(A)$ is a closed set and is the smallest closed set containing $A$.

Th (Ensuring our definitions of open and closed correspond to the ones topologically): A set $U$ is open iff $\mathbb{R}\setminus U$ is closed. Likewise, a set $F$ is closed iff $\mathbb{R}\setminus F$ is open.

Th: The union of a finite collection of closed sets is closed

The intersection of an arbitrary collection of closed sets is closed

$\mathbb{R}$ and $\varnothing$ are closed sets.

Th: For $A\subseteq \mathbb{R}$, then $A^{\prime }$ is closed.