# A Serre Fibration that is not a Hurewicz Fibration

Given two similar definitions, it is very valuable to have a counterexample distinguishing them. Here is one case where this arises: A Hurewicz fibration $p:E\to B$ of unbased spaces is a continuous map such that for every space $X$ and every commutative diagram
$$\begin{matrix} X & \longrightarrow & E\\ \downarrow & ~ & \downarrow \\ X\times I & \longrightarrow & B\end{matrix}$$
there exists a map $X\times I\to E$ making the resulting diagram commute. On the other hand, a Serre fibration $p:E\to B$ is a continuous map with the same property for every CW-complex $X$. From the definition it seems that there might be Serre fibrations that are not Hurewicz fibrations. In fact, there are! An example first appeared in R. Brown’s paper  in 1966. We define $E$ to be the subset of the Euclidean plane $\R^2$ defined by
$$E = \{ (x,y) : x\in [0,1], y = 1/n\text{ for some } n\in \N_{>0} \}\cup \{ (x,x-1) : x\in [0,1]\}$$
We take the base space $B$ to be $B = [0,1]$, and the map $p:E\to B$ is defined by $p(x,y) = x$. The fiber is then the set of poins in $[0,1]$ of the form $1/n$ or $0$. Here is a picture of this situation: (See the bottom of the post for Tikz code). This is a Serre fibration, since the image of any map $[0,1]^n\to E$ actually has to land in a connected component of $E$ (and this is the reason for the diagonal tail).

However, the composed map $F\to B$ is just the constant map $1$. The homotopy of this map with the constant map 0 cannot factor through $E$. Indeed, if there were such a factorisation, i.e. a homotopy $F:F\times I\to E$ then the composed map $F\to F\times I\to E$ where $F\to F\times I$ is given by $x\mapsto (x,1)$ would map $F$ a compact set to a noncompact set (shown projected into the vertical line).

 Brown, R. Two examples in homotopy theory. Proc. Cambridge Philos. Soc. 62 1966 575–576. MR0205252

Here is the LaTeX code for the diagram (needs \usepackage{tikz} in the preamble):

\begin{tikzpicture}[scale=2.5]
%%draw fiber
\foreach \x in {1,2,...,60}	{
\node at (1,1.5 + 1/\x) [circle,draw=blue,fill=blue,scale=0.1] {};
}
\node at (1,1.5) [circle,draw=red,fill=red,scale=0.2] {};

\draw[->] (0.5, 1.4) -- (0.5,1.1)
node [right,midway] {$i$}
;
\node at (1.3,2) {$F$};

%%We draw the space E
\foreach \x in {1,2,...,60}	{
\node at (1,1/\x) [circle,draw=blue,fill=blue,scale=0.1] {};
}
\foreach \x in {1,2,...,60}	{
\draw (0,1/\x) -- (1,1/\x);
}

\draw (0,-1) -- (1,0);
\node at (1,0) (red point) [circle,draw=red,fill=red,scale=0.2] {};
\node at (1.3,0) {$E$};

%%The base space
\draw (0,-1.5) -- (1,-1.5);
\node at (1,-1.5) [circle,draw=red,fill=red,scale=0.2] {};
\node at (1.3,-1.5) {$B$};

%%arrows
\draw[->] (0.5, -1) -- (0.5,-1.3)
node [right,midway] {$p$}
;

%%left projection illustration
\draw (-0.5,1) -- (-0.5,-1);
\foreach \x in {1,2,...,60} {
\node at (-0.5,1/\x) [circle,draw=blue,fill=blue,scale=0.1] {};
}
\node at (-0.5,-1) [circle,draw=red,fill=red,scale=0.2] { };
\end{tikzpicture}