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 real projective space $\mathbb{R}P^n$ is the projective space of the real vector space $\mathbb{R}^{n+1}$.
Equivalently this is the Grassmannian $Gr_1(\mathbb{R}^{n+1})$.
Every continuous map $\mathbb{R}P^n\rightarrow\mathbb{R}P^n$ for $n$ even has a fixed point. This does not hold for $n$ odd as in this case the continuous map $\mathbb{R}P^n\rightarrow\mathbb{R}P^n, [x_0:x_1:\ldots:x_{n-1}:x_n]\mapsto[x_1:-x_0:\ldots:x_n:-x_{n-1}]$ does not have a fixed point.
(Hatcher 02, page 155 exercise 2)
For $n\neq 1,3,7$, the smallest number $k$, so that there exists an immersion of real projective space $\mathbb{R}P^n$ into cartesian space $\mathbb{R}^{k-1}$, is exactly the topological 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
for the topological complexity (with convention $\operatorname{TC}(*)=1$).
(Farber & Tabachnikov & Yuzvinsky 02, Proposition 18)
(CW-complex structure)
For $n \in \mathbb{N}$, the real projective space $\mathbb{R}P^n$ admits the structure of a CW-complex.
Use that $\mathbb{R}P^n \simeq S^n/(\mathbb{Z}/2)$ is the quotient space of the Euclidean n-sphere by the $\mathbb{Z}/2$-action which identifies antipodal points.
The standard CW-complex structure of $S^n$ realizes it via two $k$-cells for all $k \in \{0, \cdots, n\}$, such that this $\mathbb{Z}/2$-action restricts to a homeomorphism between the two $k$-cells for each $k$. Thus $\mathbb{R}P^n$ has a CW-complex structure with a single $k$-cell for all $k \in \{0,\cdots, n\}$.
(homotopy groups of real projective space)
The homotopy groups of real projective plane can be calculated with the long exact sequence of homotopy groups (Hatcher 02, Theorem 4.41.) of the fiber bundle $S^0\rightarrow S^n\rightarrow\mathbb{R}P^n$ and are given by
(homology and cohomology of real projective space)
The ordinary homology groups of real projective space $\mathbb{R}P^n$ can be calculated using its CW structure and are given by
Similarly the ordinary cohomology groups of $\mathbb{R}P^n$ are
One has $H_{n-1}(\mathbb{R}P^n)\cong\mathbb{Z}_2$ für $n$ odd and $H_{n-1}(\mathbb{R}P^n)\cong 1$ for $n$ even, hence $\mathbb{R}P^n$ is orientable iff $n$ is odd.
The infinite real projective space $\mathbb{R}P^\infty \coloneqq \underset{\longrightarrow}{\lim}_n \mathbb{R}P^n$ is the classifying space $B O(1)$ for real line bundles. It has the homotopy type of the Eilenberg-MacLane space $K(\mathbb{Z}/2,1) = B \mathbb{Z}/2$.
See also:
On topological complexity of real projective space and connection with their immersion:
Homotopy groups, homology and cohomology of real projective space:
Computation of cohomotopy-sets of real projective space:
Last revised on March 12, 2024 at 02:50:52. See the history of this page for a list of all contributions to it.