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 it it satisfies the right lifting property with respect to maps of the form
for all topological spaces $X$, where $I$ denotes the topological interval.
This right lifting property is in this context 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 spaces 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 model structure on topological spaces and 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 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).
(covering spaces are Hurewicz fibrations)
Every covering space projection is a Hurewicz fibration, by this prop..
The historical paper of Hurewicz is
A decent review of Hurewicz fibrations, Hurewicz connections and related issues is in
A textbook account of the homotopy lifting property is for instance in
See also
R. Schwänzl, R. Vogt, Strong cofibrations and fibrations in enriched categories, 2002.
the textbooks on algebraic topology by Whitehead and Spanier.
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