Property Suggestion: Has a universal covering space

[felixpernegger]

Property Suggestion

A space is said to be Has a universal cover (abbrev: HUC) provided it is the empty space or has a nonempty simply connected cover.

Note:

We have to be careful regarding the empty space here annoyingly, since the "empty cover" exists...

There is also the closely related notion of an initial cover which is usually equivalent to universal covers and more theoretical in nature I'd say.
I think its better to universal covers instead of initial covers for two reasons:

• Universal covers do appear to be a bit more common (Geoffrey gave an overview in a past issue somewhere)

• Because "simply connected" is a well-studied topological property, my hopes are much higher finding theorems for pi-base about universal covers than initial ones. (see the theorems below)

Rationale

This property is important in algebraic topology. Using universal covers, ome can often calculate the fundamental group.
A discussion of this topic is likely found in every introductory AT book.

Relationship to other properties

HUC => Path connected

For locally path connected spaces, HUC iff Semilocally simply connected

Simply connected => HUC

Exluding spaces we dont know to be semilocally simply connected, path connected etc., these theorems already cover all but 1 space if im not mistaken.

A very relevant MSE post is the following:
https://math.stackexchange.com/questions/4329030/necessary-and-sufficient-conditions-for-x-to-have-a-universal-covering-space

We are mostly done with locally simply connected I think. Cellular spaces and the ones about higher homotopy groups are more complicated.
So I would add this property next or locally contractible.

[GeoffreySangston]

I absolutely support this. I have a feeling we should add the three variants on locally contractible first. I'll make an issue for that right now.

[prabau]

Yes, locally contractible first.

[prabau]

To me the more usual "universal covering space" instead of "universal cover" seems preferable, more understandable.
(Somehow, "cover" makes me think of cover of a space by open sets, which has nothing to do with this.)

[mathmaster13]

HUC always implies SLSC.
As for the empty set issue, we decided to call the empty set simply connected. Also, covering maps (from what I've seen) are surjections, so "nonempty" is implied if the base space is nonempty.

Also, "has an initial cover" is implied by HUC for nice spaces (at least for spaces which are both path connected and locally path connected), which motivates the name universal cover.

[felixpernegger]

Also, covering maps (from what I've seen) are surjections, so "nonempty" is implied if the base space is nonempty.

not in every definition I have seen

[GeoffreySangston]

covering maps are surjections

This is not a universal standard. I always emphasize 'surjective covering map'. The definitions in Hatcher (p. 29) and Spanier (p. 62) do not imply a covering map is surjective, for example. These particular sources define a covering space projection as a map $p : X \to Y$ such that every point in $Y$ has an evenly covered open neighborhood. An open $V$ is evenly covered if $p^{-1}(V)$ is a disjoint union of sheets ${U_\alpha}$ (terminology used in Hatcher and elsewhere), which are open set but not necessarily connected, such that $p$ restricts to a homeomorphism $U_\alpha \to V$ for each $\alpha$. If $p^{-1}(V)$ is empty, then this condition holds vacuously. Hence an open set $V$ not in the image of $p$ is vacuously evenly covered.

Note the definition implies covering space projections are open maps, so the image is open. Consider $y$ not in the image of $p$. Then $y$ has an evenly covered open neighborhood $U$. Suppose $p^{-1}(U)$ is nonempty, hence there exists a sheet $U_0$ of $p^{-1}(U)$. Then $p$ restricts to a homeomorphism $U_0 \to U$. Since $y$ is not in the image of this map, this results in a contradiction. Hence $p^{-1}(U)$ is empty. It follows that the complement of the image is open, so the image is closed. Hence if the codomain is connected and the domain is nonempty, then the covering map is surjective, but this is not necessary in general. Note that in this conception a map $X \to Y$ may be a covering map onto a proper open subspace of connected $Y$, but not be a covering map. E.g., any inclusion map of a proper open subset of connected $Y$ satisfies this. So the definition depends essentially on the codomain.

With this definition, the canonical map from the empty space to any space is a covering map. This reminded me that when talking about the special property of having an initial cover, we really only care about an initial nonempty connected cover.

Munkres on the other hand requires covering projections to be surjective by definition. A different flavor of the definition of covering space appears in Steenrod's book where he defines them as special types of bundles, and Steenrod requires bundle projections to surject.