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
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
A generalization of the Tietze extension theorem to equivariant functions provides conditions under which a continuous and equivariant function from a subspace of a topological G-space to another topological G-space has an extension to a continuous and equivariant function to the full $G$-space.
(Tietze-Gleason extension theorem)
Let
$G$ be a compact Lie group,
$X$ a topological G-space,
$E$ be a finite-dimensional orthogonal $G$-linear representation, regarded as a Euclidean topological G-space,
$A \overset{f}{\longrightarrow} E$ a $G$-equivariant continuous function.
If
then $f$ has an extension to an equivariant continuous function $\widehat f$ on all of $X$.
(Gleason 50, see Palais 60, Theorem 1.4.3)
Other/more general conditions for the equivariant extension to exist:
(Jaworowski-extension theorem)
If
the ambient domain G-space $X$ is a
with a finite number of orbit types;
the domain $A \subset X$ is a
such that for every $G$-orbit type $(H)$ in the complement $X \setminus A$
the fixed locus $E^H$ is an absolute neighbourhood retract.
Then every continuous function $f \colon A \to E$ has an extension to a $G$-equivariant continuous function $\widehat f$ on an open neighbourhood $A \subset O_A \subset X$
Moreover, if the above fixed loci $E^H$ are even absolute retracts, then an extension $\widehat f$ exists on all of $X$:
extension theorems | continuous functions | smooth functions |
---|---|---|
plain functions | Tietze extension theorem | Whitney extension theorem |
equivariant functions | equivariant Tietze extension theorem |
Andrew Gleason, Spaces with a compact Lie group of transformations, Proc. Amer. Math. Soc. 1 (1950), 35-43 (doi:10.1090/S0002-9939-1950-0033830-7)
Richard Palais, Theorem 1.4.3 in: The classification of $G$-spaces, Memoirs of the American Mathematical Society, Number 36, 1960 (ISBN:978-0-8218-9979-3 pdf, pdf)
Jan Jaworowski, Equivariant extensions of maps, Pacific J. Math. Volume 45, Number 1 (1973), 229-244 (euclid:1102947720)
Jan Jaworowski, Extending equivariant maps for compact Lie group actions, Bull. Amer. Math. Soc. Volume 79, Number 4 (1973), 698-701 (euclid:1183534741)
Jan Jaworowski, Extensions of $G$-maps and Euclidean $G$-retracts, Math Z (1976) 146: 143 (doi:10.1007/BF01187702)
Jan Jaworowski, An equivariant extension theorem and $G$-retracts with a finite structure, Manuscripta Math (1981) 35: 323 (doi:10.1007/BF01263266)
Richard Lashof, The Equivariant Extension Theorem, Proceedings of the American Mathematical Society Vol. 83, No. 1 (Sep., 1981), pp. 138-140 (jstor:2043909)
Last revised on September 22, 2021 at 15:45:23. See the history of this page for a list of all contributions to it.