CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
For $n \in \mathbb{N}$ a natural number, the $n$-dimensional ball or $n$-disk in $\mathbb{R}^n$ is the topological space
equipped with the induced topology as a subspace of the Cartesian space $\mathbb{R}^n$.
Its interior is the open $n$-ball
Its boundary is the $(n-1)$-sphere.
More generally, for $(X,d)$ a metric space then an open ball in $X$ is a subset of the form
for $x \in X$ and $r \in (0,\infty) \subset \mathbb{R}$. (The collection of all open balls in $X$ form the basis of the metric topology on $X$.)
There are also combinatorial notions of disks. For instance that due to (Joyal), as entering the definition of the Theta-category. See for instance (Makkai-Zawadowski).
A simple result on the homeomorphism type of closed balls is the following:
A compact convex subset $D$ in $\mathbb{R}^n$ with nonempty interior is homeomorphic to $D^n$.
Without loss of generality we may suppose the origin is an interior point of $D$. We claim that the map $\phi: v \mapsto v/{\|v\|}$ maps the boundary $\partial D$ homeomorphically onto $S^{n-1}$. By convexity, $D$ is homeomorphic to the cone on $\partial D$, and therefore to the cone on $S^{n-1}$ which is $D^n$.
The claim reduces to the following three steps.
The restricted map $\phi: \partial D \to S^{n-1}$ is continuous.
It’s surjective: $D$ contains a ball $B = B_{\varepsilon}(0)$ in its interior, and for each $x \in B$, the positive ray through $x$ intersects $D$ in a bounded half-open line segment. For the extreme point $v$ on this line segment, $\phi(v) = \phi(x)$. Thus every unit vector $u \in S^{n-1}$ is of the form $\phi(v)$ for some extreme point $v \in D$, and such extreme points lie in $\partial D$.
It’s injective: for this we need to show that if $v, w \in \partial D$ are distinct points, then neither is a positive multiple of the other. Supposing otherwise, we have $w = t v$ for $t \gt 1$, say. Let $B$ be a ball inside $D$ containing $0$; then the convex hull of $\{w\} \cup B$ is contained in $D$ and contains $v$ as an interior point, contradiction.
So the unit vector map, being a continuous bijection $\partial D \to S^{n-1}$ between compact Hausdorff spaces, is a homeomorphism.
Any compact convex set $D$ of $\mathbb{R}^n$ is homeomorphic to a disk.
$D$ has nonempty interior relative to its affine span which is some $k$-plane, and then $D$ is homeomorphic to $D^k$ by the theorem.
Open balls are a little less rigid than closed balls, in that one can more easily manipulate them within the smooth category:
The open $n$-ball is homeomorphic and even diffeomorphic to the Cartesian space $\mathbb{R}^n$
For instance, the smooth map
has smooth inverse
This probe from ${\mathbb{R}}^n$ witnesses the property that the open $n$-ball is a (smooth) manifold. Hence, each (smooth) $n$-dimensional manifold is locally isomorphic to both ${\mathbb{R}}^n$ and $\mathbb{B}^n$.
From general existence results about smooth structures on Cartesian spaces we have that
In dimension $d \in \mathbb{N}$ for $d \neq 4$ we have:
every open subset of $\mathbb{R}^d$ which is homeomorphic to $\mathbb{B}^d$ is also diffeomorphic to it.
See the first page of (Ozols) for a list of references.
In dimension 4 the analog statement fails due to the existence of exotic smooth structures on $\mathbb{R}^4$. See De Michelis-Freedman.
Let $C \subset \mathbb{R}^n$ be a star-shaped open subset of a Cartesian space. Then $C$ is diffeomorphic to $\mathbb{R}^n$.
This is a folk theorem. But explicit proofs in the literature are very hard to find. See the discussion at References. An explicit proof has been written out by Stefan Born, and this appears as the proof of theorem 237 in (Ferus).
Suppose $T$ is a star-shaped open subset of ${\mathbb {R}}^n$ centered at the origin. Theorem 2.29 in Lee proves that there is a function $f$ on ${\mathbb{R}}^n$ such that $f\gt 0$ on $T$ and $f$ vanishes on the complement of $T$. By applying bump functions we can assume that $f\le 1$ everywhere and $f=1$ in an open $\epsilon$-neighborhood of the origin; by rescaling the ambient space we can assume $\epsilon=2$.
The smooth vector field $V\colon x\mapsto f(x)\cdot x/{\|x\|}$ is defined on the complement of the origin in $T$. Multiply $V$ by a smooth bump function $0\le b\le 1$ such that $b=1$ for ${\|x\|} \gt 1/2$ and $b=0$ in a neighborhood of 0. The new vector field $V$ extends smoothly to the origin and defines a smooth global flow $F\colon \mathbb{R} \times T\to T$. (The parameter of the flow is all of $\mathbb{R}$ and not just some interval $(-\infty,A)$ because the norm of $V$ is bounded by 1.) Observe that for $1/2\lt {\|x\|} \lt 2$ the vector field $V$ equals $x\mapsto x/{\|x\|}$. Also, all flow lines of $V$ are radial rays.
Now define the flow map $p\colon{\mathbb{R}}^n_{\gt 1/2}\to T_{\gt 1/2}$ as $x\mapsto F({\|x\|}-1, \frac{x}{{\|x\|}})$ for ${\|x\|} \gt 1/2$. (The subscript $\gt 1/2$ removes the closed ball of radius $1/2$.) The flow map is the composition of two diffeomorphisms,
hence itself is a diffeomorphism. (Note particularly that the latter map is surjective. In detail: a flow line is a smooth map of the form $L: (A,B) \to T$, where $A$ and $B$ can be finite or infinite. If $B$ is finite and the limit of $L(t)$ as $t \to B$ exists, then the vector field $V$ vanishes at $B$. In our case $V$ can only vanish at the boundary of $T$, which is precisely what we want for surjectivity.)
Finally, define the desired diffeomorphism $d\colon{\mathbb{R}}^n\to T$ as the gluing of the identity map for ${\|x\|} \lt 2$ and as $p$ for ${\|x\|}\gt 1/2$. The map $g$ is smooth because for $1/2\lt {\|x\|} \lt 2$ both definitions give the same value.
Let $I(\Delta^n) \subset \mathbb{R}^n$ be the interior of the standard $n$-simplex. Then there is a diffeomorphism to $\mathbb{B}^n$ defined as follows:
Parameterize the $n$-simplex as
Then define the map $f : I(\Delta^n) \to \mathbb{R}^n$ by
(Thanks to Todd Trimble.) One way to think about it is that $I(\Delta^n)$ is the positive orthant of an open $n$-ball in $l^1$ norm, so that in the opposite direction we have a chain of invertible maps
which we simply invert to get the map $f$ above.
One central application of balls is as building blocks for coverings. See good open cover for some statements.
That an open subset $U \subseteq \mathbb{R}^4$ homeomorphic to $\mathbb{R}^4$ equipped with the smooth structure inherited as an open submanifold of $\mathbb{R}^4$ might nevertheless be non-diffeomorphic to $\mathbb{R}^4$, see
The proof that open star-shaped regions are diffeomorphic to a ball appears as theorem 237 in
It is a lengthy proof, due to Stefan Born.
A simpler version of the proof apparently appears on page 60 of
Apparently this proof is little known. For instance in a remark below lemma 10.5.5 of
it says:
It seems that open star shaped sets $U \subset M$ are always diffeomorphic to $\mathbb{R}^n$, but this is extremely difficult to prove.
And in
one finds the statement:
Actually, the assertion that an open geodesically convex set in a Riemannian manifold is diffeomorphic to $\mathbb{R}^n$ is common in literature, but it is a more subtle issue than it may seem, and references to a complete proof are hard to find (but see [Grom]).
Here “Grom” refers to
where the relevant statement is 1.4.C1 on page 8. Note however that the diffeomorphism considered there is only of $C^1$ class, not $C^\infty$, so that this is not a proof, either.
For a discussion of diffeomorphisms between geodesically convex regions and open balls see good open cover.
See also the Math Overflow discussion here.