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People suggesting that you can't make a coarser, still Hausdorff topology are wrong. Their proofs are invalid, and so is the conclusion. I construct a coarser topology that is still Hausdorff. You can think of this space as adding a point at $+\infty$ and gluing it to 0. Let $\tau$ be the usual topology of open sets on $\mathbb{R}$. Define $\tau' \subset \tau$ as the collection of sets in $\tau$ that, if they contain 0, contain some interval $(\alpha,+\infty)$.
Let $M$ be a Möbius strip, $C\subset M$ a circle running through the center, and consider the fiber bundle $M\setminus C \to C$ given by some projection. What happens if we try to construct a section in the way you describe? For each $c\in C$, we need to pick a point either above or below $C$. But as you can easily verify with some paper, $M\setminus C$ is connected! So if we "follow it through", we will find that there is no consistent choice of side.
Let $Y$ be a compact manifold, $X$ a topological space and $f: X \to Y$ a surjective map. Suppose further that every point in $Y$ has arbitrarily small open neighbourhoods such that their preimages in $X$ are contractible. Under these conditions, can we find a good cover for $Y$ such that its preimage is also a good cover for $X$ (i.e. such that any finite intersections of open sets in the cover is contractible)?
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