Let \(E\) be a subspace of \(\mathbb{R}^2\) obtained by deleting a subspace of \(\{0\} \times \mathbb{R}\). For which such spaces \(E\) is the projection \(E \to \mathbb{R}\), \((x, y) \mapsto x\), a fiber bundle?
Solution
Write \[ E=\mathbb R^2-\bigl(\{0\}\times A\bigr), \qquad p(x,y)=x. \] For \(x\ne0\), the fiber is \(\mathbb R\), while \[ p^{-1}(0)=\mathbb R-A. \] If \(p\) is a fiber bundle over the connected base \(\mathbb R\), all fibers are homeomorphic. Hence a necessary condition is \[ \mathbb R-A\cong\mathbb R. \] As a subspace of \(\mathbb R\), a space homeomorphic to \(\mathbb R\) is a connected one-manifold without boundary, hence is an open interval, allowing infinite endpoints. Thus necessarily \[ \mathbb R-A=(a,b) \] for some \[ -\infty\le a<b\le\infty. \]
This condition is also sufficient. Put \(I=(a,b)\). Away from \(0\) the projection is already the trivial product. It remains to trivialize over a small interval \((-\varepsilon,\varepsilon)\).
For \(t>0\), choose an increasing homeomorphism \[ h_t:\mathbb R\to I \] which equals the identity on the portion of \(I\) at distance at least \(t\) from each finite endpoint, compresses the left tail into \((a,a+t]\) when \(a\) is finite, and compresses the right tail into \([b-t,b)\) when \(b\) is finite. For example, at a finite left endpoint use \[ h_t(y)=a+t\exp\!\left(\frac{y-a-t}{t}\right) \qquad(y\le a+t), \] and use the identity for \(y\ge a+t\) until the analogous right-end interpolation begins; at a finite right endpoint use \[ h_t(y)=b-t\exp\!\left(-\frac{y-b+t}{t}\right) \qquad(y\ge b-t). \] For sufficiently small \(t\) the two transition regions are disjoint. If an endpoint is infinite no compression is needed there. These formulas give a jointly continuous family in \((t,y)\), with \[ h_t(y)\longrightarrow y \] for every \(y\in I\) as \(t\to0\), and points outside \(I\) are sent toward the corresponding missing endpoint.
Define \[ H:p^{-1}(({-}\varepsilon,\varepsilon))\longrightarrow (-\varepsilon,\varepsilon)\times I \] by \[ H(x,y)= \begin{cases} (x,h_{|x|}(y)),&x\ne0,\\ (0,y),&x=0. \end{cases} \] The displayed limiting behavior gives continuity at the central fiber; the same argument for \(h_t^{-1}\) gives continuity of the inverse. Thus \(H\) is a fiber-preserving homeomorphism.
Therefore \[ \boxed{p:E\to\mathbb R\text{ is a fiber bundle exactly when }\mathbb R-A\text{ is an open interval (possibly unbounded).}} \]