This is a question I found here (on pg.2 example 1.2.2): http://web.pdx.edu/~erdman/PTAC/problemtext_pdf.pdf
Definition of interior and interior point: Let $A \subseteq \mathbb{R}$. The point $a$ is an interior point on $A$ if some $\epsilon$-neighborhood of $a$ lies entirely in $A$. The set of all interior points of $A$ is denoted $A^0$ and is called the interior.
The book gives this proof: Let $a$ be an arbitrary point in $(0,1)$. Choose $\epsilon$ to be the smaller of the numbers $a$ and $1-a$. Then $J_{\epsilon}(a)=(a-\epsilon, a+\epsilon) \subseteq (0,1)$ (because $\epsilon \le a$ implies $a-\epsilon \ge 0$, and $\epsilon \le 1-a$ implies $a+\epsilon \le 1$).
My question. In the reasoning noted in parenthesis at the end, shouldn't those inequalities be strict inequalities rather than having the "or equals" part? My example is if $a=0.5$. You would not want $\epsilon=0.5$ since that would results in that in touching the bounds 0 and 1, which is not part of our interval. Does this make sense?
If $a=0.5$ then $\varepsilon=\min(a,\,1-a)=0.5$ and then the neighborhood of $a$ is $(a-\varepsilon,\,a+\varepsilon)\ =\ (0,1)$, does not contain its endpoints.
Of course, you can also choose any $\varepsilon$ that is smaller or equal than $\min(a,\,1-a)$.