quaternionic projective space$\,\mathbb{H}P^1$
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, β-Chern-Weil theory
Cartan geometry (super, higher)
The Reeb sphere theorem in differential topology says that:
If a compact differentiable manifold $X$ admits a differentiable function $X \to \mathbb{R}$ with exactly two critical points which are non-degenerate, then $X$ is homeomorphic to an n-sphere (with its Euclidean metric topology).
In fact this holds true even if the two critical points happen to be degenerate (Milnor 64, theorem 1β on p. 166)
Let $x_{min}$ be the critical point at which $f$ attains its minimum and $f_{max}$ the critical point at which it attains its maximum.
By the fact that $X$ is a differential manifold we find an closed neighbourhood $B \subset X$ of $x_{min}$ which is diffeomorphic to an closed ball.
By the assumption that there are no other critical points, $f$ is monotonically increasing along the flow lines $\gamma$ of $\nabla f$ on $X \setminus \{x_{min}, x_{max}\}$, in that the function $\nabla_f(d \gamma_t) = \nabla f(\nabla f) = {\vert \nabla f\vert^2 }$ is strictly positive:
Hence for $C \subset X$ a compact, the extreme value theorem implies that this function attains its minimum.
This implies that for every $y \in \mathbb{R}$ with $f(x_{min}) \lt y \lt f(x_{max})$ there exists $t \in \mathbb{R}$ such that the flow $\phi_{\nabla f}$ along $\nabla f$ satisfies $\phi_{\nabla f}(x,t) \geq y$ for all $x \in \partial B$. In particular this is the case for $y$ the maximum of $f|_{C}$. This implies that $C$ is contained in the image of $B$ under some flow of $\nabla f$. But this image is, by the nature of the flow, diffeomorphic to a Euclidean ball.
In conclusion this shows that every compact subspace of $X$ is contained in an open subspace diffeomorphic to a Euclidean ball. With this the claim follows by the Brown-Stallings lemma.
Notice that the Reeb sphere theorem does not speak of diffeomorphism to an n-sphere, just about homeomorphism. Indeed, the theorem may be used as an ingredient in the construction of exotic smooth structures on some $n$-spheres.
John Milnor, theorem 4.1 in Morse theory (pdf)
John Milnor, Differential topology, chapter 6 in T. L. Saaty (ed.) Lectures On Modern Mathematic II 1964 (web, pdf)
Last revised on December 1, 2019 at 14:16:04. See the history of this page for a list of all contributions to it.