Let $[a,b]$ be any finite closed interval.
Am I correct to say that the interior of this set is $[a,b]$? Since the interior of a set are all the points in the set in which we can construct an open ball centered at that point which is completely contained inside the set - then for example an open ball centered at $b$ would be completely contained in $(a,b]$, since with respect to $[a,b]$, no part of the ball exists outside this interval?
Similarly, is the closure of $[a,b)$ here, $[a,b)$? Since although any open ball centered at $b$ has intersection with $[a,b)$ - the point $b$ does not exist in the space?