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
Serre fibration$\Leftarrow$ Hurewicz fibration $\Rightarrow$ Dold fibration $\Leftarrow$ shrinkable map
A continuous map $p \;\colon\; E\longrightarrow B$ of topological space is called a Hurewicz fibration if it satisfies the right lifting property against all continuous functions
including any topological space $X$ as one end of the cylinder over it (where $I \coloneqq [0,1]$ denotes the topological interval).
In this context, the defining right lifting property is called the homotopy lifting property, because the maps from $X\times I$ are understood as homotopies. In more detail, for every space $X$, any homotopy $F:X\times I\to B$, and a continuous map $f:X\to E$, there is a homotopy $\tilde{F}:X\times I\to E$ such that $f =\tilde{F} \circ\sigma_0 :=\tilde{F}_0$ and $F=p\circ\tilde{F}$:
Strictly speaking, the “all” in this context should be interpreted to refer to all spaces in whatever ambient category of TopologicalSpaces one is working in, since frequently this is a convenient category of spaces. In theory, therefore, a map in such a category could be a Hurewicz fibration in that category without necessarily being a Hurewicz fibration in the category of all topological spaces, but in practice this usually doesn’t make a whole lot of difference.
Instead of checking the homotopy lifting property, one can instead solve a universal problem:
A map is a Hurewicz fibration precisely if it admits a Hurewicz connection. (See there for details.)
There is a Quillen model category structure on Top where fibrations are Hurewicz fibrations, cofibrations are closed Hurewicz cofibrations and weak equivalences are homotopy equivalences – see at model structure on topological spaces and specifically at Strøm's model category.
There is a version of Hurewicz fibrations for pointed spaces, as well as in the slice category $Top/B_0$ where $B_0$ is a fixed base (the slice model structure).
There is also a model category whose fibrations are the Hurewicz fibrations and whose weak equivalences are the weak homotopy equivalences, obtained by mixing the above model structure with the classical model structure on topological spaces.
Every Hurewicz fibration is a Serre fibration. Conversely, a Serre fibration between CW-complexes is a Hurewicz fibration.
The following proposition says that being a Hurewicz fibration is a local property with respect to the codomain space, at least as long as the local open cover used is numerable.
(Local recognition of Hurewicz fibrations over numerable open covers)
Let $p \colon E \to B$ be a continuous function such that there exists a numerable open cover $U \overset{\phi}{\longrightarrow} B$ over which $E$ is a Hurewicz fibration, hence such that the pullback $\phi^\ast p$ is a Hurewicz fibration.
Then $p$ is a Hurewicz fibration
A proof may be found spelled out in e.g. May 99, Sec. 7.4
The concept of Hurewicz fibrations makes sense also more generally in the presence of a (well behaved) interval object, see for instance the early example of such in (Kamps 72) and see (Williamson 13) for review and further developments. Discussion with a view towards homotopy type theory is in (Warren 08).
(empty bundles are Hurewicz fibrations)
All empty bundles $\varnothing \longrightarrow B$ are Hurewicz fibrations, because none of the commuting squares that one would have to non-trivially lift in actually exist:
since the empty topological space is a strict initial object: There is no morphism to it from any inhabited space $X$
(If $X$ itself is empty, then so is $X \times I$ and then a square only exists if $B$ in turn is empty. This square, consisting entirely of empty spaces, does have a unique lift, trivially, and hence also the empty bundle over the empty space is a Hurewicz fibration – for what it’s worth.)
(covering spaces are Hurewicz fibrations)
Every covering space projection is a Hurewicz fibration, by this prop..
(numerable fiber bundles are Hurewicz fibrations)
Every numerable fiber bundle, hence in particular every fiber bundle over a paracompact topological space, is a Serre fibration.
By definition of local triviality, for a numerable fiber bundle $p \colon E \to B$ – such as a fiber bundle over a paracompact topological space – there exists a numerable open cover $\phi \colon U \to B$ such that the pullback bundle $\phi^\ast p$ is trivial, i.e. is the Cartesian product projection
(for $F$ the typical fiber). Since such a projection is clearly a Hurewicz fibration the claim follows by the local recognition of Hurewicz fibrations over numerable open covers, from Prop. .
Notice that the example of empty bundles (Example ) is a special case of a fiber bundle: With typical fiber the empty topological space.
The original article:
A decent review of Hurewicz fibrations, Hurewicz connections and related issues is in
Textbook accounts:
Tammo tom Dieck, Klaus Heiner Kamps, Dieter Puppe, Chapter II of: Homotopietheorie, Lecture Notes in Mathematics 157 Springer 1970 (doi:10.1007/BFb0059721)
Alan Hatcher, from p. 61 on in Algebraic Topology (web)
See also
R. Schwänzl, R. Vogt, Strong cofibrations and fibrations in enriched categories, 2002.
the textbooks on algebraic topology by Whitehead and Spanier.
Peter May, A concise course in algebraic topology, University of Chicago Press 1999 (ISBN: 9780226511832, pdf)
Tyrone Cutler, Fibrations III – Locally trivial maps and bundles, 2020 (pdf, pdf)
Abstract analogues of Hurewicz fibrations can be found in
summarised in
and further developed in
Discussion with an eye towards homotopy type theory is in
Last revised on October 21, 2021 at 14:51:40. See the history of this page for a list of all contributions to it.