Why must a topology on a set contain the empty set?

I have just had my first week of topology, and I have a question that is rather basic.

Why must the empty set be an element of any given topology?

(For reference, the definition of a topology T I am working with, for a set X:

1. X and the empty set must be in T.
2. the union of elements of any subset in T, is also in T.
3. the intersection of elements of any finite subcollection in T, is also in T.)

Solution 1:

It's more convenient this way. $\def\less{\smallsetminus}$

Usually in math when you have definitions that involve the empty set, the reason they do is that it's more elegant, convenient, economical to phrase things this way.

Suppose you didn't want to deal with the empty set. You could define topology to be a collection of nonempty subsets of X that satisfy a few axioms. But then your axioms would have to be more complicated and include special cases.

You'd have to say that only a nonempty finite intersection of open sets is open. You could modify the 3rd axiom to say that. But the reason you need that axiom in the first place is that very frequently in proofs you take a few open sets, and rely on the fact that their intersection is still open (and therefore has some qualities useful to you). In the new version, every time you need to intersect two open sets in a proof, you'd have to list two cases: a) if they're disjoint, then their intersection is empty and therefore... b) if they're not disjoint, then their intersection is open and therefore... Your proofs would be longer and more tedious.

Similarly, normally you define a subset $Y$ of $X$ to be closed if $X \less Y$ is open. Then under the usual definition the whole space $X$ is a closed set. There're many reasons why you want that to be the case and why that corresponds to the intuition of "closed set". But if you don't consider the empty set to be open by definition, you don't get $X$ to be closed. Again, you can patch it up by modifying the definition of closed set: "A subset $Y$ of $X$ is closed if $Y=X$ or $Y$ is the complement of an open set". This will work. But, again, your proofs will become more complicated: when you have a closed set $Y$ about which you know little, you can't just assume anymore that $X \less Y$ is open and work with that, now you have to consider two cases.

These are just two examples, but there're more. Basically, it turns out that while making the empty set open by definition may not look a priori very natural to you, it makes all kinds of objects and properties "click together" more naturally and economically than otherwise.

Solution 2:

Note that the fact that $X$ and $\emptyset$ are open can be seen as a consequence of the two other axioms if one allows the empty intersection and the empty union, where $X$ is the empty intersection (that is, the intersection over the empty family) and $\emptyset$ is the empty union (that is, the union over the empty family) .

Seeing things this way is useful not only in this aspect. For instance, every collection of subsets of a given set is a subbasis for some topology (in the sense that the unions of finite intersections is a topology) if we take this approach, instead of having to require that the collection covers the space.

Solution 3:

One reason the empty set is open is due to vacuous truth, which is a reason that has nothing to do with topology at all, but rather logic.

The implication $p \implies q$ is always true if the statement $p$ is false (regardless of statement $q$ -- statement $q$ can be any random statement). This is called vacuous truth.

Now, one definition or characterization of openness is that $A \subseteq X$ is open if for each $x \in A$, we can find an open neighborhood $U$ with $x \in U$ and $U \subseteq A$.

In other words, $A \subseteq X$ is open if for every $x$, the statement $$x \in A \implies \exists U \text{ open with } x \in U \subseteq A $$ is true.

Let's check the truth value of this statement if $A = \emptyset$. For $\emptyset$ to be open, we need for every $x$ that the following statement is true: $$x \in \emptyset \implies \exists U \text{ open with } x\in U \subseteq \emptyset $$

Now, for every $x$, the "if" statement in the implication above, which is the statement $x \in \emptyset$, is always false. There is absolutely no $x$ satisfying $x \in \emptyset$, so $x \in \emptyset$ is false. But if $p$ is false, then for any statement $q$, $p \implies q$ is true (this is vacuous truth). Thus for every $x$, the above implication holds, so by definition of open, $\emptyset$ is open.

Thus, to recap, the empty set $\emptyset$ is open because it satisfies the definition of being open. But it satisfies this definition vacuously.

So, in short, one reason $\emptyset$ is open has nothing to do with topology at all, but with logic and why we take the statement $p \implies q$ to be true no matter what statement $q$ is as long as statement $p$ is false.

(Extra non-topological example of vacuous truth: "If $1 + 2 = 20$, then $5 = 16$" is a true statement (vacuously) since the "if" part is false.)

Solution 4:

Two important concepts in topology are interior and closure of a set $A\subseteq X$. (Interior is the largest open subset, closure is the smallest closed super-set.) If $\emptyset$ would not be open, then interiors and closures would be undefined in general. Indeed, if the full space $X$ is not closed (in itself), what would be the closure $\overline{X}$?