Find the interior points for the following sets

Find the interior points for the following sets:

  1. The set $mathbb{Q}$ in $mathbb{R}$
  2. The set $B = {(x,y): 0 < x+y leq 1 }$ in $mathbb{R}^2$

My Attempt:

  1. $A^0= emptyset$, since there is no point $qinmathbb{Q}$ within $mathbb{R}$ for which we can find a $r$-neighnorhood such that the interior contains only rationals.

  2. $B^0= {(x,y): 0 < x+y <1 }$ in $mathbb{R^2}$

Are these correct?

Answer

Your proposed answers are correct. To make your claims rigorous and complete, for the first case you need to show for example that in between any two distinct real numbers there is an irrational number, and for the second case you need to show that if $0 < x + y < 1$ then $(x,y)$ is an interior point of the set (according to the usual definition of interior points for sets in ${mathbb R}^d$), and if $x + y = 1$ then $(x,y)$ is not an interior point (in fact, it is on the boundary of the set).

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