Are $\emptyset$ and $X$ closed, open or clopen? It is indeed a very basic question but I am confused:
(1) In an 2013 MSE posting under general topology here, I was told that $\emptyset$ is an open set and therefore I assume $X$ must be open too.
(2) But in Wikipedia page on clopen set here, it says "In any topological space $X$, the $\emptyset$ and the whole space $X$ are both clopen."
(3) And yet in another Wikipedia page on closed set here, "The $\emptyset$ is closed, the whole set is closed."
I must have missed something. Can you help me with a supreme verdict, once and for all, as sure as the sun rises from the east each morning, if $X$ and $\emptyset$ are open, closed or clopen. Of course I am talking about topology, thanks for your time.

To put it simply: sets are not doors! Being open does not mean that the set is not closed, and being closed do not implies that the set is not open. Yes, the empty set and the whole space are both open and closed. Another more dramatic example is: take a metric space $X$, with the discrete metric. Then every singleton $\{a\}$ (in fact, every subset of $X$) is clopen. Balls $B(a,r)$ with radius $r < 1$ are contained in $\{a\}$, hence $\{a\}$ is open. And $\{a\}$ is also closed, because it's complement is $X \setminus\{a\} = \bigcup_{b \in X, b \neq a}\{b\}$, a union of open sets, hence open. 
