# What is a Covering Space (intuitively) and is it related to the concept of open cover?

I do know the definition of a covering space (e.g. Wikipedia: https://en.wikipedia.org/wiki/Covering_space), but am a bit confused what exactly is it.

Is it related to the concept of Cover in point set topology? (https://en.wikipedia.org/wiki/Cover_(topology))

My observation is that the covering space ($C$) is totally different from the original space ($X$), while for cover, the union of sets in the cover contain $X$.

Any intuitive (non-rigorous is ok) explanation of covering space will be greatly appreciated. Thanks!

• For what it's worth, the use of the term "covering" in both concepts is a linguistic coincidence, i.e., not mathematically significant. Commented Nov 10, 2015 at 14:23
• I think the covering space for $S^1$ (helix is covering space, projection map is covering map) is good for intuition, and is also the first concrete example most people learn. I don't really think that a cover is really related to a covering space. I think the word "covering" in covering space is just trying to express that an open neighborhood $U\subseteq X$ can be "covered" by disjoint open sets in $C$. Commented Nov 10, 2015 at 14:24
• @andrew I disagree that the use of the word "covering" in both situations is not mathematically significant! For instance in the étale topology, a covering map is a "covering" by a single "open" set. Best, Commented Nov 10, 2015 at 16:09
• @Bruno: That may be, but étale spaces are not the origin of the terminology. :) Commented Nov 10, 2015 at 16:23

What you should really be focussing on is the concept of a "covering map". It is tempting to focus on "covering space", but if you do so then you will lose both intuition and mathematical precision. There is relation between the two terminologies: given a covering map $$p : C \to X$$, the range $$X$$ is called the "base space", and the domain $$C$$ is called a "covering space of $$X$$".

Given a base space $$X$$, the intuitive idea of a covering map $$p : C \to X$$ is that $$p$$ "winds" $$C$$ around $$X$$.

For example, you can wind the circle around itself two times using the covering map $$p_2 : S^1 \to S^1$$ defined with complex variables by the formula $$p_2(z)=z^2$$. Similarly, the map $$p_3(z)=z^3$$ winds the circle around itself three times, and the map $$p_k(z) = z^k$$ winds the circle around itself $$k$$ times. Each of these is an example a covering map $$p_k :S^1 \to S^1$$ whose base space (range) is $$S^1$$, and whose covering space (domain) is also $$S^1$$.

If you've ever played with rubber bands, the 3-fold covering map $$p_3:S^1 \to S^1$$ is easy to visualize by giving the rubber band a little bit of a twist until it settles into a 3-fold wrapping picture, as seen here (and discussed in this blog by James Propp):

For another example, you can wind the real line $$\mathbb{R}$$ infinitely around $$S^1$$, using the formula $$e(s) = e^{2 \pi i s}$$. This defines a covering map $$e : \mathbb{R} \to S^1$$ whose base space is $$S^1$$ and whose covering space is $$\mathbb{R}$$. Another depiction of this covering map is the projection from the infinite helix $$\{(\cos(2 \pi t),\sin(2 \pi t), t) \, \bigm| \, t \in \mathbb{R} \}$$ onto the first two coordinates.

An important feature of a covering map $$f : C \to X$$ is that it winds evenly over the space $$X$$ (look in the Wikipedia definition, under the heading "formal definition", for the discussion of "evenly covered neighborhoods" of the base space $$X$$). One consequence of this is that $$f$$ is a $$k$$-to-one map over every point of the base space, where $$k$$ is a constant called the degree of $$f$$. The $$k$$ can be a positive integer, or $$\infty$$, or any cardinal number. In the example above, $$p_3 : S^1 \to S^1$$ is a covering map of degree $$3$$, and $$e : \mathbb{R} \to S^1$$ has degree $$\infty$$.

• Thanks! Very enlightening! Commented Nov 11, 2015 at 0:04