For a subset $E$ of a metric space $(X,d)$, one defines the closure $\overline{E}$ as the intersection of all the closed sets that contain $E$.
Let's put aside the fact that it must be intersection of closed sets for one moment. If we put aside it, then $\overline{E}$ is the $\inf$ of the sets of upper bounds of $E$.
However, in "General topology" from Willard & Stephen, definition $3.5$ the authors say "$\overline{E}$ is the smallest closed set containing E (in the sense that it is contained in every closed set containing $E$)". When I read "smallest closed set" I can't help thinking about least element of the set of upper bounds of $E$. Is that what the authors wanted to convey ?
If we say $\overline{E}$ is the $\inf$ of the sets of upper bounds of $E$ like I said in the second paragraph, then it implies that it is contained in every (closed) set containing $E$, that is, in every upper bound of $E$. However, defining $\overline{E}$ as some lower bound of the set of upper bounds of $E$ doesn't imply that it is the $\inf$ of the set of upper bounds of $E$.
So, two questions:
- why do the authors use the term smallest, when there is not much about least element in this issue ?
- why do the authors caracterize (in what I have quoted from the book) $\overline{E}$ just as some lower bound of the set of upper bounds of $E$ when, according to me, it should better be the greatest lower bound of the set of upper bounds of $E$ ?
Additional question:
- Is there any problem with my way of putting aside the closedness in my considerations ?
Thank you very much in advance!