0
$\begingroup$

Is there a standard name for the closure of a submanifold of some fixed manifold M?

Example. The closed interval [0, 1] is not a manifold, because there is no atlas which contains charts at either 0 or 1. However, it is the closure of the manifold (0, 1) in the larger manifold ℝ.

Other Examples. The unit ball in ℝn for n>0 (the above example is isomorphic to the case n=1); the unit sphere in ℝn (which is itself already a manifold, but equal to its own closure in that embedding).

These objects may either have intrinsic boundary points, as with a closed interval, or lack them as with the unit sphere. They ought to have in common that removing their intrinsic boundary points (if any) leaves you with a submanifold of M, whose closure is the original set.

Is there standard terminology for such objects?

  • 0
    I can only say that when I took a course about manifolds, we simply called the object you're referring to as a manifold with a boundary. So I tend to believe that there is no standard terminology for such objects. But i'm not sure.2011-10-07
  • 1
    The problem with "manifold with boundary" is that some of these don't have boundaries.2011-10-07
  • 0
    @Niel: the usual definitions of manifold with boundary makes it a superset of manifolds: the set of boundary points is usually allowed to be empty. If you want to emphasize it you can say they are "manifolds (possibly) with boundary". Also, are you working in the topological or smooth category? In the latter, your set may be in fact a manifold with corners, and not a manifold with boundary.2011-10-07
  • 0
    Also, how do you define the "intrinsic boundary points" of a closed subset of a topological space?2011-10-07
  • 0
    @WillieWong: the 'intrinsic boundary points' are perhaps not a topological concept, but a concept from (unsuccessful attempts to define) an atlas on a space. An intrinsic boundary point is one that cannot be covered by a chart, or at least cannot be mapped to an interior point of a neighborhood of a Euclidean space by a chart.2011-10-07
  • 0
    @WillieWong: this particular sort of object occurred to me while light-heartedly contemplating fundamentals of physics. Event-space is a 4-manifold; reference frames are atlases; an object is a set of events forming a (closure of a) submanifold of event-space, which can be smoothly parameterized thus and so; etcetera. So it would probably be the "smooth category". My physicsy bias makes the term "with boundary" unappealing, however, unless it actually has 'intrinsic' boundary points. (I would like to use terminology consistent with the usual parlance that the cosmos probably has no boundary.)2011-10-07

1 Answers 1

2

In the most generality, you probably can't say anything besides "the closure of a submanifold". Consider the following construction. Take the polar coordinate system in $\mathbb{R}^2$ by $(r,\theta)$. Define the open sets

$$\Omega_n = \{ \theta \in (\frac{1}{n} - \frac{1}{2^n}, \frac{1}{n} + \frac{1}{2^n}); r > 0\} $$

Then

$$ \Omega = \left(\cup_{n = 1}^\infty \Omega_n\right) \cup \{ r > 1\} $$

is an open set, and so is a submanifold of $\mathbb{R}^2$ (and it is connected). Its closure necessarily includes the origin. But at the origin $\bar{\Omega}$ is quite a horrible set; as is at the entire half-line given by $\theta = 0$. I'm personally not aware of any established terminology for classes of objects which include $\bar{\Omega}$.


Note also that the unit sphere can be considered to be the closure of itself. It can also be considered to be the closure of the submanifold given by the unit sphere with the north pole removed. So there's some question as to what the "boundary points" are if you are only presented with the closure of the set, but not the set itself.

  • 0
    The set $\bar\Omega$ is indeed an somewhat torn-up object topologically. Perhaps there is terminology for simply-connected-closures-of-submanifolds?2011-10-17
  • 0
    @Niel: what would you mean by that phrase? If you take the unit circle in $\mathbb{R}^2$ and remove the point $(1,0)$, it is a perfectly innocuous submanifold (_sans_ boundary) with codimension 1. Its closure (in the obvious way) cannot be simply connected. Another example, what if your submanifold is already not simply-connected? Then what would you mean by simply-connected-closure?2011-10-17
  • 0
    I actually meant connected in such a way that you can closed path to a point. I momentarily forgot that my original question also asked for a definition which would include the unit circle; and consequently I didn't realize that for the special case I'm asking about which omits the unit circle, "manifold with boundary" probably is the right term for any *proper* submanifold.2011-10-17