General Topology – Are ? and X Closed, Open, or Clopen?

general-topology

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.

Best Answer

They are all both open and closed.

Let me make this a bit more clear. By definition of a topology (from wikipedia):

A topological space is then a set $X$ together with a collection of subsets of $X$, called open sets and satisfying the following axioms:

  • The empty set and $X$ itself are open.
  • Any union of open sets is open.
  • The intersection of any finite number of open sets is open.

So just from the definition itself it follows that $∅$ and $X$ are open.

Furthermore a set is closed (by definition) if the complement is open. Therefore $∅$ and $X$ are closed (they are each others complement).

The term clopen means that a set is both open and closed, so they are both also clopen.

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