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.

A minimal requirement on any topological space $(X,\tau)$ is that both $\varnothing$ and $X$ be open sets. By the definition of closed sets, these requirements imply that $\varnothing^c=X$ and $X^c=\varnothing$ are always closed.
To sum up, in any topological space, the empty set and the whole set are always both open and closed, hence clopen.
Your question “are $\varnothing$ and $X$ closed, open or clopen” is basically six questions in one:
(1) Is $\varnothing$ closed? Answer: yes.
(2) Is $\varnothing$ open? Answer: yes.
(3) Is $\varnothing$ clopen? Answer: yes.
(4) Is $X$ closed? Answer: yes.
(5) Is $X$ open? Answer: yes.
(6) Is $X$ clopen? Answer: yes.
