My general topology textbook just gave the definition of euclidean topology on $\mathbb{R}$ but unfortunately did not provide any examples and I was hopping that someone here could help me with some questions I have. The definition they gave is the following:
A subset $S$ of $\mathbb{R}$ is said to be open in the euclidean topology on $\mathbb{R}$ if it has the following property:
(1)$\ \ \ \ $For each $x \in S$, there exists $a,b\in\mathbb{R}$, with $a<b$, such that $x \in ]a,b[\subseteq S$.
My questions are the following:
Let $A=[1,2]$ then we can define an interval $]1-\epsilon,2+\epsilon[$ and then we would have that $[1,2]\subset ]1-\epsilon,2+\epsilon[$, wouldn't that mean that $[1,2]$ is opened?
Let $A=]1,2[\ \cup\ ]3,4[$,
we can define a new interval $]1,4[$ such that $\forall x\in A,x\in ]1,4[$. But it is not true that $]1,4[\subseteq A$. Rather we have that $A \subseteq ]1,4[$. So the set $A$ does not have property (1) thus not being opened.
On the other hand, $]1,2[$ and $]3,4[$ are opened and the union of open sets is also opened. Since $A =]1,2[\ \cup\ ]3,4[$ then $A$ must be opened. So is $A$ opened or not?
The set $[1,2]$ is not open, because there are no numbers $a$ and $b$, with $a<b$, such that $1\in(a,b)$ and that $(a,b)\subset[1,2]$.
And $(1,2)\cup(3,4)$ is open. For instance, if $x\in(1,2)$ and $r=\min\{|x-1|,|x-2|\}$, then $(x-r,x+r)\subset(1,2)\subset(1,2)\cup(3,4)$.