# The empty space

## Definition

The empty space is the topological space with no points. That is, it is the empty set equipped with its unique topology.

## Properties

The empty space is the initial object in Top. It satisfies all separation, compactness, and countability conditions (separability, first countability, second-countability). It is also both discrete and indiscrete, a distinction it shares only with the point.

### Connectedness

Debate rages over whether the empty space is connected (and also path-connected). With the common naive definitions that “a space is connected if it cannot be partitioned into disjoint nonempty open subsets” and “a space is path-connected if any two points in it can be joined by a path,” the empty space is trivially both connected and path-connected.

However, in some ways these definitions are too naive. The question of whether the empty set is (path-)connected is analogous in many ways to the question of whether $1$ is prime. The above definitions are then analogous to saying that “a natural number $p$ is prime if any factor of it is either equal to $1$ or to $p$,” according to which $1$ is prime—but there are better definitions that exclude $1$.

For instance, we may say that “$p$ is prime if it has exactly two factors, itself and $1$;” with this definition $1$ is not prime, since it has exactly one factor. Likewise, we may say that a space is (path-)connected if it has exactly one (path-)component?; with this definition the empty space is not connected, since it has exactly zero components. (Lest you question that last statement, note that the correct definition of a (path-)component of a space $X$ is as an equivalence class of points of $X$ under some equivalence relation. There is a unique equivalence relation on the empty set, and it has zero equivalence classes.)

Here are some other reasons why the empty space should not be considered (path-)connected:

• If the empty space were (path-)connected, unique decomposition into (path-)connected components would fail: $X \cup Y = \emptyset \cup X \cup Y = \dots$. This is analogous to how if $1$ were a prime, then unique factorization into primes would fail: $6 = 2 \cdot 3 = 1 \cdot 2 \cdot 3 = 1 \cdot 1 \cdot 2 \cdot 3 = \dots$.

• In homotopy theory, one defines a space $X$ to be $k$-connected if $\pi_i(X)$ is trivial (that is, has exactly one element) for $i \le k$. When $k =0$ this says that $\pi_0(X)$ should have exactly one component—that is, that $X$ should be path-connected. (Actually, this definition really only makes sense if we phrase it in terms of homotopy groupoids; homotopy groups are only defined once we choose a basepoint, which is clearly impossible for the empty space.)

• Category-theoretically, one may say that a space $X$ is connected if the functor $hom(X,-)$ preserves coproducts. Since $\hom(\emptyset,-)$ is constant at the point, it certainly does not preserve coproducts.

See too simple to be simple for general theory.

Revised on January 10, 2013 02:48:58 by Toby Bartels (64.89.53.241)