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
A metrisable topological space $Y$ is an absolute neighborhood retract (ANR) if (Borsuk 32, p. 222) for any embedding $Y \subset Z$ as a closed subspace in a metrisable topological space $Z$, $Y$ is a neighborhood retract of $Z$.
A metrisable topological space $Y$ is an absolute retract if for any embedding $Y\subset Z$ as a closed subspace in a metrisable topological space $Z$, $Y$ is a retract of $Z$.
(ANR is a local property for metrizable spaces)
A metrizable topological space which admits an open cover by absolute neighbourhood retracts is itself an absolute neighbourhood retract.
(review in Hu 65, III Thm. 8.1)
A metrisable topological space is an absolute retract precisely if it is
Every (finite-dimensional) metrizable locally Euclidean topological space – in particular every topological manifold – is an absolute neighbourhood retract.
By Prop. (see also Hu 65, III Cor. 8.3).
In fact:
Every paracompact Banach manifold is an absolute neighbourhood retract.
Let $X$ be an absolute neighbourhood retract (ANR) and $A \xhookrightarrow{i} X$ a closed subspace-inclusion. Then $A$ is an ANR precisely iff the inclusion $i$ is a Hurewicz cofibration.
Every finite-dimensional locally finite CW-complex is an absolute neighbourhood retract.
The notion of absolute neighbourhood retract is due to
Further development:
Olof Hanner, Some theorems on absolute neighbourhood retracts, Arkiv För Matematik Band 1 nr 30 (1950) (doi:10.1007/BF02591376)
James Dugundji, Note on CW polytopes, Portugaliae mathematica (1952) 11 1 (1952) 7-10-b (dml:114693)
Yukihiro Kodama, Note on an absolute neighborhood extensor for metric spaces, Journal of the Mathematical Society of Japan 8 3 (1956) 206-215 (doi:10.2969/jmsj/00830206)
Karol Borsuk, Concerning the classification of topological spaces from the stand point of the theory of retracts, Fundamenta Mathematicae 46 (3) (1959) 321-330 (dml:213516)
Discussion for infinite-dimensional manifolds:
Textbook accounts and review:
Karol Borsuk, Theory of retracts, Vol. 44. Państwowe Wydawn. Naukowe, 1967
Sze-Tsen Hu, Theory of Retracts, Wayne State University Press (1965) (google-books)
Sibe Mardešić, Absolute Neighborhood Retracts and Shape Theory (pdf)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Def. 4.2.10 in: Algebraic topology from a homotopical viewpoint, Springer (2002) (doi:10.1007/b97586, toc pdf)
See also:
Last revised on September 19, 2021 at 02:30:18. See the history of this page for a list of all contributions to it.