9

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.

Community
  • 1
A.Magnus
  • 3,527

4 Answers4

12

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.

Loreno Heer
  • 4,460
6

By the first axiom in the definition of a topology, $X$ and $\emptyset$ are open. However, closed sets are precisely those whose complements are open, by definition. Hence the empty set and $X$, being each others complements, are also closed. So, they are clopen.

user141592
  • 6,118
6

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.

Ivo Terek
  • 77,665
  • Ask you a question, I hope I can say it clear: Let $B \subset A$ and $B$ is open. Absence of any other instructions, $A$ has to be closed since it is $B$'s complement. Now if $A$ is clopen, then its neutrality does not have any bearing on open-ness or the closed-ness of $B$. Am I correct? – A.Magnus Feb 01 '15 at 01:54
  • But it is not true that $A$ is the complement of $B$.. – Ivo Terek Feb 01 '15 at 01:55
  • I made typo and just corrected it. Sorry for confusion. – A.Magnus Feb 01 '15 at 01:56
3

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.

triple_sec
  • 23,377