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
(see also Chern-Weil theory, parameterized homotopy theory)
For $V$ a vector space and $r$ a cardinal number (generally taken to be a natural number), the Grassmannian $Gr(r,V)$ is the space of all $r$-dimensional linear subspaces of $V$.
For $n \in \mathbb{N}$, write $O(n)$ for the orthogonal group acting on $\mathbb{R}^n$. More generally, say that a Euclidean vector space $V$ is an inner product space such that there exists a linear isometry $V \overset{\simeq}{\to} \mathbb{R}^n$ to $\mathbb{R}^n$ equipped with its canonical inner product. Then write $O(V)$ for the group of linear isometric automorphisms of $V$. For the following we regard these groups as topological groups in the canonical way.
For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th Grassmannian of $\mathbb{R}^k$ is the coset topological space.
where the action of the product group is via its canonical embedding $O(n)\times O(k-n) \hookrightarrow O(n)$.
Generally, for $W \subset V$ an inclusion of Euclidean vector spaces, then
where $V-W$ denotes the orthogonal complement of $W$ in $V$.
The group $O(k)$ acts transitively on the set of $n$-dimensional linear subspaces, and given any such, then its stabilizer subgroup in $O(k)$ is isomorphic to $O(n)\times O(k-n)$. In this way the underlying set of $Gr_n(k)$ is in natural bijection to the set of $n$-dimensional linear subspaces in $\mathbb{R}^k$. The realization as a coset as above serves to euqip this set naturally with a topological space.
Similarly, the real Stiefel manifold is the coset
The quotient projection
is an $O(n)$-principal bundle, with associated bundle $V_n(k)\times_{O(n)} \mathbb{R}^n$ a vector bundle of rank $n$. In the limit (colimit) that $k \to \infty$ is this gives a presentation of the $O(n)$-universal principal bundle and of the universal vector bundle of rank $n$, respectively.. The base space $Gr_n(\infty)\simeq_{whe} B O(n)$ is the classifying space for $O(n)$-principal bundles and rank $n$ vector bundles.
The real Grassmannians $Gr_n(\mathbb{R}^k)$ and the complex Grassmannians $Gr_n(\mathbb{C}^k)$ admit the structure of CW-complexes. Moreover the canonical inclusions
are subcomplex incusion (hence relative cell complex inclusions).
Accordingly there is an induced CW-complex structure on the classifying space
A proof is spelled out in (Hatcher, section 1.2 (pages 31-34)).
The Grassmannian $Gr_r(\mathbb{R}^{n+1})$ (def. ) is
for $r = 0$: the point;
for $r = 1$: the real projective space $\mathbb{R}P^n$;
for $r = n+1$: the point;
for $r \gt n+1$: the empty space $\emptyset$.
If $V$ is an inner product space, then the orthogonal complement defines an isomorphism between $Gr(r,V)$ and $Gr(\dim V - r,V)$.
Textbook accounts include
Stanley Kochmann, section 1.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Allen Hatcher, section 1.2 of Vector bundles & K-theory (web)
SL Kleiman, Geometry on Grassmannians and applications to splitting bundles and smoothing cycles, Publications Mathématiques de l’IHÉS, 1969, numdam/pdf
Lecture notes include
Last revised on January 28, 2021 at 06:33:28. See the history of this page for a list of all contributions to it.