algebraic-topologycovering-spacesgeneral-topology
The Wikipedia page about covering spaces asserts that:
The universal cover (of the space $X$) covers any connected cover (of the space $X$).
I'm not able to find a proof of this result. Any idea how to find it or how to prove it?
Here the definition of universal cover is just that of a covering being simply connected. Moreover, a part from connectedness of the other cover there is no other hypothesis.
Suppose $p:\overline{C}\to C$ is a universal covering. By definition, around every point in $C$ is an open set that lifts up to $\overline{C}$. So, locally, $\overline{C}$ looks just like $C$. Suppose one wanted to cut up $C$ into small (contractible) patches and then stitch them together again to form $\overline{C}$ - the problem is that $\overline{C}$ is to be simply connected, so if (say) we started stitching patches around a nontrivial loop in $C$ when we wrap back around to the basepoint we can't stitch that last patch back to the original patch, instead we have to create a copy of the original patch and continue on from there.
Consider the space $C=\Bbb C\setminus\{0\}$. If one takes a counterclockwise loop from $-1$ around $0$ back to itself, then the last patch cannot be stitched to the first, so we should make a copy of the original patch to stitch it to. In the picture below, we've literally lifted the copy above the original:
$\hskip 2in$ 
If we continue this process indefinitely, then there will lots of copies of pieces of $C$ that are being stitched together. Given a point in $C$ in a patch, there will be many copies of that patch in our quilt, and so many lifts of that point - what allows us to distinguish between lifts of the same point is how we got to it from the original basepoint. Thus, we can interpret points in $\overline{C}$ as points in the original space $C$ but with a "memory" of how we got there from a basepoint.
This inspires us to formalize our construction by letting elements of $\overline{C}$ be paths in $C$, modulo endpoint-fixing homotopy. Points in $\overline{C}$ are specified by points in $C$ with a memory of how we got there from the basepoint, so if we got to $x\in C$ via a path $\gamma$ in $C$ and $U$ is any basic nbhd of $x\in C$, then the lift $\overline{U}$ of that nbhd is comprised of points $\overline{u}$, and to specify these $\overline{u}$ we must say which points of $C$ they are (done: they lie above $U$) and how we got to them. We got to these points in $\overline{U}$ by first travelling along $\gamma$ from the basepoint to $x$ and then wiggled around within $U$ itself.
As for your other question, try lifting the paths. Say that $D\to C$ is a covering, where $D$ is a familiar space you know well, and in particular you know $D$ is simply connected. Our construction of $\overline{C}$ is comprised of paths emanating from (say) $x\in C$. To see what the corresponding point of $D$ is, just lift the path from $C$ to $D$ and look at its endpoint! This is the isomorphism.
Here is a picture of the universal cover of a figure eight I made for a topology course in 1991 (!), using Macintosh-Pascal. Note the self-similarity of the "ends".
Best Answer
Sofie Verbeek's answer is perfect, but let me go back to the definition of a universal covering space. There are two different approaches: Some (perhaps most) authors define a universal covering $p : E \to X$ of a connected space $X$ to be one such that $E$ is simply connected, other authors define it by the property that it covers any covering $p' : E' \to X$ with a connected $E'$.
In my opinion the second definition is the better one because it is explains the name. If you accept this point of view, then it is a theorem that a simply connected covering of a connected and locally path connected $X$ is a universal covering.
A nice reference is Chapter 2 Section 5 of
Spanier, Edwin H. Algebraic topology. Vol. 55. No. 1. Springer Science & Business Media, 1989.
It contains a number of interesting results and examples (e.g. of a non-simply connected universal covering space of a of a connected and locally path connected space).