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
    @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
    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