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
Theorems
Given a continuous map $\pi : E\to B$ of topological spaces, one constructs the mapping cocylinder $Cocyl(\pi)$ as the pullback
where $\mathcal{P}(B)$ is the path space in Top, the space of continuous paths $u:[0,1]\to B$ in $B$, and where $\mathcal{P}(B)\to B$ is the map sending a path $u$ to its value $u(0)$. The cocylinder can be realized as a subspace of $E\times \mathcal{P}(B)$ consisting of pairs $(e,u)$ where $e\in E$ and $u:[0,1]\to B$ are such that $\pi(e)=u(0)$.
A Hurewicz connection is any continuous section
of the map $\pi_!:\mathcal{P}(E)\to Cocyl(\pi)$ given by $\pi_!(u)=(u(0),\pi\circ u)$.
A map $\pi:E\to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for $\pi_!$.
To see that consider the following diagram
where $pr_E: Cocyl(\pi)\to E$ is the restriction of the projection $E\times B^I\to E$ to the factor $E$ and the map $Cocyl(\pi)\times I\to B$ is the evaluation $(e,u,t)\mapsto u(t)$ for $(e,u)\in Cocyl(\pi)$. The right-hand square is commutative and this square defines a homotopy lifting problem. If $\pi$ is a cofibration this universal homotopy lifting problem has a solution, say $\tilde{s}:Cocyl(p)\times I\to E$. By the hom-mapping space adjunction (exponential law) this map corresponds to some map $s:Cocyl(\pi)\to \mathcal{P}(E)$. One can easily check that this map is a section of $\pi_!$.
Conversely, let a Hurewicz connection $s$ exist, and fill the right-hand square of the diagram with diagonal $\tilde{s}$ obtained by hom-mapping space adjunction. Let the data for the general homotopy lifting problem be given: $\tilde{f}:Y\to E$, $F:Y\times I\to B$ with $F_0 = p\circ \tilde{f}:Y\to E$; let furthermore $F':Y\to \mathcal{P}(B)$ be the map obtained from $F$ by the hom-mapping space adjunction. By the universal property of the cocylinder (as a pullback), there is a unique mapping $\theta: Y\to Cocyl(\pi)$ such that $pr_{\mathcal{P}(B)}\circ\theta=F':Y\to \mathcal{P}(B)$ and $pr_E\circ\theta =\tilde{f}:Y\to E$. Now notice that the by composing the horizontal lines we obtain $\tilde{f}$ upstairs and $F$ downstairs, hence the external square is the square giving the homotopy lifting problem for this pair. The lifting is then given by $\tilde{s}\circ (\theta\times id_I):Y\times I\to E$. Simple checking finishes the proof.
Of course there are many other equivalent characterizations of Hurewicz fibrations.
If $\pi:E\to B$ is a covering space where $B$ is Hausdorff, then $\pi_!$ is a homeomorphism; thus in that case the Hurewicz connection is unique.
If $\pi$ is a smooth principal bundle equipped with a distribution of horizontal spaces forming an Ehresmann connection, then one can define a corresponding “smooth” Hurewicz connection in the sense that the Ehresmann connection provides a continuous choice of smooth path lifting, with prescribed initial point, of a smooth path in the base. This can be expressed in terms as a continuous section of $\pi_!^{smooth}:\mathcal{P}^{smooth}(E)\to Cocyl^{smooth}(\pi)$ where the subspaces of smooth paths are used.
The original article is
A review is for instance in