I need a little help understanding exactly what an interior & boundary point are/how to determine the interior points of a set. If you could help me understand why these are the correct answers or also give some more examples that would be great. Thanks~
a. $\{1/n\colon n\in \!\, \mathbb{N} \!\,\}$
interior= $\varnothing \!\,$
The reason that S has no interior points is that for each of its points 1/n, any open set containing 1n contains points that are not of the form 1/n.
boundary point=
b. $[0,3]\cup \!\,(3,5)$
interior= $(0,5)$
boundary point=
c.${r\in \!\,\mathbb{Q} \!\,:0<r<\sqrt2}$
interior= $\varnothing \!\,$
boundary point=
d. ${r\in \!\,\mathbb{Q} \!\ :r \ge\ \sqrt2}$
interior= $\varnothing \!\,$
boundary point=
e.$[0,2]\cap \!\,[2,4]$
interior= $\varnothing \!\,$
The reason that S has no interior points is that the intersection of [0,2] and [2,4] is 2, and for the point 2, any open set that contains 2 will contain points that are outside of the set.
boundary point=
Best Answer
Your stated reason for (a) is mistaken. You said
But this is confused. Note that the given set (call it $S$) is $\left\{\frac1n\mid n\in \Bbb N\right\}$. This is not the same as $\left\{\frac1n\mid \frac1n\in \Bbb N\right\}$. The latter would be the set $\{1\}$.
The reason that $S$ has no interior points is that for each of its points $\frac1n$, any open set containing $\frac1n$ contains points that are not of the form $\frac1n$. For example, $\frac12$ is not an interior point because any open set containing $\frac12$ must also contain some of the points that are between $\frac12$ and $\frac13$, which are not included in $S$.
Your other answers for the interiors are correct, although perhaps not for the right reasons. (You didn't give any.)