It is well-known that a fiber bundle under some mild hypothesis is a fibration, but I don't know any examples of fiber bundles which aren't (Hurewicz) fibrations (they should be weird examples, I think, because if the base space is paracompact then the bundle is a fibration).


Does anybody know an example?


Thanks!



Example of fiber bundle that is not a fibration
$\newcommand{\RR}{\mathbb{R}}
\newcommand{\To}{\longrightarrow}
\newcommand{\id}{\mathrm{id}}$The example described in [Tom Goodwillie's answer](https://mathoverflow.net/questions/106497/non-trivial-vector-bundle-over-non-paracompact-contractible-space/106563#106563) to a related mathoverflow question essentially solves this question. Specifically, Tom defines an orientable, non-trivial, real line bundle $L$ over a contractible space $X$. The principal $GL\_1^+(\RR)$-bundle — $GL\_1(\RR)$ will work as well — associated to $L$ is then a locally trivial fibre bundle $p:E\to X$. The principal bundle $p:E\to X$ does not admit a section, since the line bundle $L$ is not trivial. It follows that $p$ cannot be a Hurewicz fibration: otherwise it would admit a section, given that $X$ is contractible. For convenience, I will give below the construction of $p:E\to X$ and a few details of the proof — mostly copied from Tom's answer, and following his notation where possible. Please upvote Tom Goodwillie's answer, which is certainly shorter and more readable.


### The base space $X$


Let $X$ be the space obtained by gluing two copies of $\RR$ along $\RR^+$:
$$ X = (\RR\times\{0,1\})\mathbin{/}{\sim} $$
where ${\sim}$ is generated by $(x,0)\sim(x,1)$ for $x\in\RR^+$. This space is not Hausdorff, and is closely related to the well-known *line with two origins*. Let $q:\RR\times\{0,1\}\to X$ be the quotient map. Define two open subspaces covering $X$ by $U=q(\RR\times\{0\})$ and $V=q(\RR\times\{1\})$. Finally, let $g:X\to\RR$ be the continuous function determined by $g(q(x,i))=x$, and define
$$f=g|\_{U\cap V}: U\cap V \To \RR^+$$
Importantly, observe that $f$ does not extend to a continuous map $X\to\RR^+$.


### The total space $E$


Consider the topological abelian group $G=GL\_1^+(\RR)=(\RR^+,\cdot)$ given by the positive reals with multiplication. Let $E\_U=U\times G$ and $E\_V= V\times G$ denote the trivial $G$-bundles over $U$ and $V$, respectively. Construct the principal $G$-bundle $E$ over $X$ by gluing $E\_U$ and $E\_V$ along $U\cap V$ via the $G$-isomorphism
$$ \varphi\_f : E\_U|\_{U\cap V}\overset{\simeq}{\To} E\_V|\_{U\cap V} $$
defined by
$$ \varphi\_f(x,g)=\bigl(x,f(x)\cdot g\bigr) $$
More concretely, $E$ is obtained from $E\_U \amalg E\_V$ by identifying $(x,g)\in E\_U$ with $\varphi\_f(x,g)\in E\_V$ for each $x\in U\cap V$. As in Tom Goodwillie's answer, we could just as well use any other continuous function $f:U\cap V\to\RR^+$ which does not extend to a continuous function $X\to\RR^+$.


### Non-triviality of the principal bundle $E\to X$


The projection map $p:E\to X$ gives a principal $G$-bundle over $X$, which comes with canonical isomorphisms $E|\_U = E\_U$ and $E|\_V = E\_V$. We will now show $p$ does not admit a section. By the construction of $E$, a section of $p:E\to X$ determines:


* a section of $E\_U=U\times G\to U$, and therefore a map $s\_U:U\to G=\RR^+$;
* similarly, a map $s\_V:V\to G=\RR^+$;
* these maps verify $s\_V(x)=f(x)\cdot s\_U(x)$ for each $x\in U\cap V$.


In particular, $f(x)=s\_V(x)/s\_U(x)$ for all $x\in U\cap V$. However, this implies that $f$ extends to a continuous function $\overline{f}:X\to\RR^+$ given by
$$ \overline{f}(x)=\frac{s\_V\bigl(q(g(x),1)\bigr)}{s\_U\bigl(q(g(x),0)\bigr)} $$
which contradicts the known non-extension property of $f$.


### Conclusion


The projection $p:E\to X$ gives a locally trivial principal $G$-bundle over $X$, since $E|\_U\simeq E\_U$ and $E|\_V\simeq E\_V$ are trivial $G$-bundles. Thus, it remains to show that $p$ is not a Hurewicz fibration. Note that $X$ is contractible. So if $p:E\to X$ were a fibration, it would necessarily admit a section. In detail, let $H:X\times I\to X$ be a null-homotopy of $\id\_X$. We can obviously lift the constant map $H\_1$ to $E$. Assuming $p$ is a Hurewicz fibration, the homotopy lifting property then produces a lift $\widetilde{H}: X\times I\to E$ of $H$ to $E$, and consequently a section $\widetilde{H}\_0$ of $p$. But we showed in the previous paragraph that $p$ admits no sections. In conclusion, $p$ is not a Hurewicz fibration.


