Connectedness and Discrete-Valued Maps

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 is disconnected if there exists a disjoint pair of open subsets and of whose union is , and that is connected otherwise. A discrete-valued map is a continuous map from a topological space to a discrete space , 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 is connected if and only if every discrete-valued map on is constant.

The proof is quite simple. If is connected, then the preimage of an element in the image of is nonempty, open, and closed: therefore, must be the whole space. Conversely, if is not connected, then we can find a disjoint pair of open subsets and of whose union is , whence the map which is 0 on and 1 on is a nonconstant discrete-valued map on .

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 by quotienting out by its connected components. The resulting space is a discrete space, and the quotient map is a surjective continuous map that is constant if and only if is a singleton. Of course, this happens precisely when has one connected component, i.e., if 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 is a connected space, a continuous map, and any discrete-valued map, then the composition must be constant, and so 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 must be constant on each connected set, and the value of must be the same because they all share a point: it follows that is constant on the union. Another illustrative example is as follows:

Proposition 4. If is a connected subset of a topological space , and if is a subset of such that , then is connected.

Again, the proof is very short. Any discrete-valued map on is a continuous, hence sequentially continuous, map that is constant on , whence we conclude that the value of on any limit point of is the same as the value takes on . It follows that is constant on , hence on .