General Topology | Mark H. Kim

I found a neat characterization of connectedness in Bredon’s Topology and Geometry: so nice, in fact, that I am compelled to write up a quick note about it.

Let us recall that a topological space $X$ is disconnected if there exists a disjoint pair of open subsets $U$ and $V$ of $X$ whose union is $X$ , and that $X$ is connected otherwise. A discrete-valued map is a continuous map $d:X \to D$ from a topological space $X$ to a discrete space $D$ , which is a topological space in which every subset is open. An alternate characterization of connectedness is as follows:

Proposition 1 (Alternate characterization of connectedness). A topological space $X$ is connected if and only if every discrete-valued map $d:X \to D$ on $X$ is constant.

The proof is quite simple. If $X$ is connected, then the preimage $d^{-1}(y)$ of an element $y$ in the image of $d$ is nonempty, open, and closed: therefore, $d^{-1}(y)$ must be the whole space. Conversely, if $X$ is not connected, then we can find a disjoint pair of open subsets $U$ and $V$ of $X$ whose union is $X$ , whence the map $d:X \to \{0,1\}$ which is 0 on $U$ and 1 on $V$ is a nonconstant discrete-valued map on $X$ .

Why would anyone think of such an alternate definition? Glen Wilson offered the following perspective, which I very much like. First off, following the “categorical” way of thinking, we submit that describing a collection of object in terms of maps between them—as opposed to the objects themselves—is a good thing. But why constant maps? We can consider “labeling” each connected component in a topological space $X$ by quotienting $X$ out by its connected components. The resulting space $Y$ is a discrete space, and the quotient map $\pi:X \to Y$ is a surjective continuous map that is constant if and only if $Y$ is a singleton. Of course, this happens precisely when $X$ has one connected component, i.e., if $X$ is connected.

This alternate characterization leads us to swift proofs of the key properties of connected space. Let us first consider the following

Proposition 2. The continuous image of a connected space is connected.

Here is a one-liner for the proof: If $X$ is a connected space, $f:X \to Y$ a continuous map, and $d:f(X) \to D$ any discrete-valued map, then the composition $d \circ f:X \to D$ must be constant, and so $f(X)$ is connected. The following triumph of intuition also admits a devastatingly simple proof:

Proposition 3. If a collection of connected sets share a point, then the union is connected.

Here, any discrete-valued map $d$ must be constant on each connected set, and the value of $d$ must be the same because they all share a point: it follows that $d$ is constant on the union. Another illustrative example is as follows:

Proposition 4. If $A$ is a connected subset of a topological space $X$ , and if $B$ is a subset of $X$ such that $A \subseteq B \subseteq \bar{A}$ , then $B$ is connected.

Again, the proof is very short. Any discrete-valued map $d$ on $X$ is a continuous, hence sequentially continuous, map that is constant on $A$ , whence we conclude that the value of $d$ on any limit point of $A$ is the same as the value $d$ takes on $A$ . It follows that $d$ is constant on $\bar{A}$ , hence on $B$ .