One theorem turns out to be very important when we’re dealing with connected spaces, or even just with a connected component of a space. If is a continuous map from a connected space to any topological space , then the image is connected. Similarly, if is path-connected then its image is path-connected.
The path-connected version is actually more straightforward. Let’s say that we pick points and in . Then there must exist and with and . By path-connectedness there is a function with and , and so and . Thus the composite function is a path from to .
Now for the connected version. Let’s say that is disconnected. Then we can write it as the disjoint union of two nonempty closed sets and by putting some connected components in the one and some in the other. Taking complements we see that both of these sets are also open. Then we can consider their preimages and , whose union is since every point in lands in either or .
By the continuity of , each of these preimages is open. Seeing as each is the complement of the other, they must also both be closed. And neither one can be empty because some points in land in each of and . Thus we have a nontrivial clopen set in , contradicting the assumption that it’s connected. Thus the image must have been connected, as was to be shown.
From this theorem we see that the image of any connected component under a continuous map must land entirely within a connected component of the range of . For example, any map from a connected space to a totally disconnected space (one where each point is a connected component) must be constant.
When we specialize to real-valued functions, this theorem gets simple. Notice that a connected subset of is just an interval. It may contain one or both endpoints, and it may stretch off to infinity in one or both directions, but that’s about all the variation we’ve got. So if is a connected space then the image of a continuous function is an interval.
An immediate corollary to this fact is the intermediate value theorem. Given a connected space , a continuous real-valued function , and points with and (without loss of generality, ), then for any there is a so that . That is, a continuous function takes all the values between any two values it takes. In particular, if is itself an interval in we get back the old intermediate value theorem from calculus.