manifolds and cobordisms
cobordism theory, Introduction
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
It may superficially seem that every locally Euclidean space is a Hausdorff topological space, since Euclidean space is. While the analogous conclusion is in fact true for genuine “local” properties of Euclidean space (such as the $T_1$-separation axiom, sobriety and local compactness, see this prop.), it is actually wrong for the “non-local” $T_2$= Hausdorff-conditon: There are locally Euclidean spaces which are not Hausdorff topological spaces, such as the line with two origins.
Since a (paracompact/sigma-compact) Hausdorff locally Euclidean space is called a topological manifold, (paracompact/sigma-compact) non-Hausdorff locally Euclidean spaces are sometimes referred to as “non-Hausdorff manifolds” (an instance of the red herring principle).
The usual example is the line with two origins: the real line with the point $0$ ‘doubled’. Explicitly, this is a quotient space of $\mathbb{R} \times \{a,b\}$ (given the product topology and with $\{a,b\}$ given the discrete topology) by the equivalence relation generated by identifying $(x,a)$ with $(y,b)$ iff $x = y \ne 0$.
Mathieu Baillif, Alexandre Gabard, Manifolds: Hausdorffness versus homogeneity (arXiv:0609098)
Steven L. Kent, Roy A. Mimna, and Jamal K. Tartir, A Note on Topological Properties of Non-Hausdorff Manifolds, (web)
Last revised on May 18, 2017 at 10:11:17. See the history of this page for a list of all contributions to it.