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
The topological complexity [Farber 2001] of a topological space is a topological invariant related to the problem of motion planning and the immersion problem of real projective space.
For a topological space $X$ its topological complexity $TC(X)$ is the smallest number $n$, so that there is an open cover $(U_k)_{k=1}^n$ of $X\times X$ by $n$ open subsets addmitting local sections $s_k\colon X\times X\rightarrow P X$ of the evaluation map
of the path space $P X$.
It is also possible to define it using the Schwarz genus? of the path space fibration $\Omega X\hookrightarrow P X\rightarrow X$.
There is also the convention of using the smallest number $n$, so that an open cover $(U_k)_{k=0}^n$ of $n+1$ open sets of $X\times X$ with the above property exists. This lowers all topological complexities by one, hence the convention used can be given by giving the topological complexity of the set with one point. ($TC(*)=1$ for the upper convention and $TC(*)=0$ for the lower convention.)
A topological space $X$ is contractible iff $TC(X)=1$.
The topological complexity is only dependent on the homotopy type of a topological space.
The topological complexity of a sphere is
(with convention $\operatorname{TC}(*)=1$)
This theorem can be generalized:
The topological complexity of a product of spheres is
(with convention $\operatorname{TC}(*)=1$).
A special case of this proposition is $TC(T^n)=n+1$ for the topological complexity of the torus.
For $n\neq 1,3,7$, the smallest natural number $k\in\mathbb{N}$, so that there exists an immersion of real projective space $\mathbb{R}P^n$ into euclidean space $\mathbb{R}^{k-1}$ is the topologial complexity $\operatorname{TC}(\mathbb{R}P^n)$ (with convention $\operatorname{TC}(*)=1$).
(Farber & Tabachnikov & Yuzvinsky 02, Theorem 12)
For $n=1,3,7$, one has $\operatorname{TC}(\mathbb{R}P^n)=n+1$ (with convention $\operatorname{TC}(*)=1$).
(Farber & Tabachnikov & Yuzvinsky 02, Proposition 18)
For any $n\in\mathbb{N}$, one has $\operatorname{TC}(\mathbb{C}P^n)=2n+1$ (with convention $\operatorname{TC}(*)=1$).
(Farber & Tabachnikov & Yuzvinsky 02, Corollary 2)
The topological complexity of a $\Sigma$ surface is
(with convention $TC(*)=1$)
For $n \geq 2$ and $g \geq 2$ one has
for the connected sum of real projective space (with convention $TC(*)=0$).
(Cohen & Vandembrouq 18, Theorem 1.3.)
The topological complexity of the Klein bottle is $4$ (with convention $TC(*)=0$).
(Cohen & Vandembrouq 16, Theorem 1)
The topological complexity of a configuration space is
(with convention $TC(*)=1$).
(Farber & Grant 08, Theorem 1)
Definition and basic properties of topological complexity:
See also:
On topological complexity of real projective space and connection with their immersion into cartesian space:
On topological complexity of connected sums:
On topological complexity of the Klein bottle:
On topological complexity of configuration space:
Last revised on February 14, 2024 at 07:49:21. See the history of this page for a list of all contributions to it.