see also algebraic topology, functional analysis and homotopy theory
topological space (see also locale)
fiber space, space attachment
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
subsets are closed in a closed subspace precisely if they are closed in the ambient space
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
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
The Hopf fibration (named after Heinz Hopf) is a canonical nontrivial circle bundle over the 2-sphere whose total space is the 3-sphere.
This we discuss below in
More generally, there are four Hopf fibrations, on the 1-sphere, the 3-sphere, the 7-sphere and the 15-sphere, respectively. This we discuss in
The Eilenberg-MacLane space $K(\mathbb{Z},2) \simeq B S^1$ is the classifying space for circle group principal bundles. By its very nature, it has a single nontrivial homotopy group, the second, and this is isomorphic to the group of integers
This means that there is, up to homotopy, a canonical (up to sign), continuous map from the 2-sphere
such that $[\phi] \in \pi_2(K(\mathbb{Z},2)) = \pm 1 \in \mathbb{Z}$.
As any map into $K(\mathbb{Z},2)$ this classifies a circle group principal bundle over its domain. This is the Hopf fibration, fitting into the long fiber sequence
In other words, the Hopf fibration is the $U(1)$-bundle with unit first Chern class on $S^2$.
An explicit topological space presenting the Hopf fibration may be obtained as follows. Identify
and
Then the continuous function $S^3 \to S^2$ defined by
gives the Hopf fibration. (Thus, the Hopf fibration is a circle bundle naturally associated with the canonical line bundle.) Alternatively, if we use
and identify this presentation of the 2-sphere with the complex projective line via stereographic projection, the Hopf fibration is identified with the map $S^3 \to S^2$ given by sending
Regard $S^1 = U(1)$ as equipped with its circle group structure. This makes $S^1$ in particular an H-space. The Hopf fibration $S^1 \to S^3 \to S^2$ is the Hopf construction applied to this H-space.
For each of the normed division algebras over $\mathbb{R}$, the real numbers, complex numbers, quaternions, octonions
there is a corresponding Hopf fibration of Hopf invariant one.
The total space of the fibration is the space of pairs $(\alpha, \beta) \in A^2$ of unit norm: ${|\alpha|}^2 + {|\beta|}^2 = 1$. This gives spheres of dimension 1, 3, 7, and 15 respectively. The base space of the fibration is projective 1-space $\mathbb{P}^1(A)$, giving spheres of dimension 1, 2, 4, and 8, respectively. In each case, the Hopf fibration is the map
($n = 1, 2, 3, 4$) which sends the pair $(\alpha, \beta)$ to $\alpha/\beta$.
When $X$ is a sphere that is an $H$-space, namely, one of the groups $S^0 = 1$ the trivial group, $S^1 = \mathbb{Z}/2$ the group of order 2, the 3-sphere special unitary group $S^3 = SU(2)$; or the 7-sphere $S^7$ with its Moufang loop structure, then the Hopf construction produces the above four Hopf fibrations:
When line bundles are regarded as models for the topological structure underlying the electromagnetic field the Hopf fibration is often called “the magnetic monopole”. We may think of the $S^2$ homotopically as being the 3-dimensional Cartesian space with origin removed $\mathbb{R}^3 - \{0\}$ and think of this as being 3-dimensional physical space with a unit point magnetic charge at the origin removed. The corresponding electromagnetic field away from the origin is given by a connection on the corresponding Hopf fibration bundle.
In complex K-theory, the Hopf fibration represents a class $H$ which generates the cohomology ring $K_U(S^2)$, and satisfying the relation $H^2 = 2 \cdot H - 1$, or $(H-1)^2 = 0$. (So in particular $H$ has an inverse $H^{-1} = 2- H$, see at Bott generator.)
A succinct formulation of Bott periodicity for complex K-theory is that for a space $X$ whose homotopy type is that of a CW-complex, we have
(It would be interesting to see whether this can be proved by internalizing the (classically easy) calculation for $K(S^2)$ to the topos of sheaves over $X$.)
The Hopf fibrations over other normed division algebras also figure in the more complicated case of real K-theory $K_O$: they can be used to provide generators for the non-zero homotopy groups $\pi_n(B O)$ for the classifying space of the stable orthogonal group, which are periodic of period 8 (not coincidentally, 8 is the dimension of the largest normed division algebra $\mathbb{O}$). [To be followed up on.]
Reviews include