So I've just started reading topology without tears and I was a bit confused at this question
Given a set $X = \{a,b,c,\}$. topology
$\tau$ =$\{X, \emptyset, \{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}\}$.
Is the point a an interior point of the set $\{a\}$
I think that it is because all points in an open set are automatically interior points but also because the set
$\{a\}$ is by definition a neighbourhood of the point as there is an open set contained inside of it namely itself is my reasoning correct?
And is it true that all points in an open set are interior points?
I'm not quite sure what the answer is.
Thanks in advance.
Best Answer
A point $x\in A$ is an interior point of $A$ if $A$ is a neighborhood of $x$. In other words, $x$ is an interior point of $A$ if there is some open set $S$ such that $x\in S$ and $S\subset A$. But then, if $A$ is open, you can just take $S=A$. So, yes, all points in an open set are interior points of that set. And so, yes, your second argument is correct. And so is the first one, by the way.