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
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Every (paracompact Hausdorff) differentiable manifold can be equipped with a Riemannian structure. Therefore the existence of a metric does not impose constraints on a topology of a manifold. The local Riemannian structure of a Riemannian manifold on the other hand can be very different with the same global topology: there are even locally many nonisometric Riemannian structures on the same manifold/neighborhood. All even dimensional manifolds allow locally a (unique up to isomorphism) symplectic structure. Symplectic manifolds, due to the Darboux theorem, are all locally isomorphic, but according to the spectacular findings of Gromov 1985, there are still global invariants of symplectic structure as well as topological constraints on even-dimensional manifolds permitting symplectic structure.
Introduction:
Further discussion“
Mikhail Gromov: Pseudo holomorphic curves in symplectic manifolds, Inventiones Mathematicae 82 2 (1985) 307–347 [pdf, doi:10.1007/BF01388806]
Dusa McDuff, Dietmar Salamon: J-holomorphic curves and symplectic topology, AMS Colloquium Publications 52 (2004) [ams:coll-52-r]
Yong-Geun Oh, Symplectic topology and Floer homology [pdf]
Last revised on February 1, 2026 at 15:08:56. See the history of this page for a list of all contributions to it.