Int(E) mean ? what does Int mean

Question

In real analysis what is Int symbol mean?

like Int(E) , int (A U B) ??

I want to know what is Int mean in real analysis

a few example will be also good

thank you

Answer 1

int $A$ means set of all interior points of $A\subseteq\mathbb{R}$

for example say $A=\mathbb{Q}$, Then int $A=\emptyset$

$A=[0,1]$, Then int $A=(0,1)$

Answer 2

$\operatorname{int}E$ is the set of all interior points in $E\subseteq\mathbb{R}$. Recall that a point $p$ is an interior point of $E$ if $B_\epsilon p\subset E$. (where $B_\epsilon p$ denotes the open ball with radius $\epsilon$, centered at $p$.) Also, note that an open set $E$ is where all the points are interior.

Answer 3

In general topology (which construction works also in metric spaces), the interior of a set $E$ is the union of all open sets contained in it. Equivalently, it is the greatest open set that is still contained in $E$. Formally, $$\operatorname{int}\,(E)=\bigcup_{\substack{U\text{ is open}\\U\subseteq E}}U.$$

Examples in the Euclidean space of the reals: \begin{align*} \operatorname{int}\,(E)=&\,E\qquad\text{for any open $E\subseteq\mathbb R$},\\ \operatorname{int}\,([0,1])=&\,(0,1),\\ \operatorname{int}\,([0,1))=&\,(0,1),\\ \operatorname{int}\,(\{x\})=&\,\varnothing\qquad\text{for any $x\in\mathbb R$},\\ \operatorname{int}\,(F)=&\,\varnothing\qquad\text{for any finite set $F\subset\mathbb R$},\\ \operatorname{int}\,(C)=&\,\varnothing\qquad\text{for any countably infinite set $C\subset\mathbb R$}. \end{align*} The last point implies, perhaps surprisingly, that if $\mathbb Q$ is the set of rational numbers, then \begin{align*} \operatorname{int}\,(\mathbb Q)=\varnothing. \end{align*} Even more surprisingly, \begin{align*} \operatorname{int}\,(\mathbb I)=\varnothing, \end{align*} where $\mathbb I$ is the set of irrational numbers.