Are continuous product projections always split? What's an example of a product projection without a continuous right inverse?
2026-03-29 04:34:20.1774758860
Are continuous product projections always split?
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I do not know what you mean by "product projection", so I will give two examples.
The hairy ball theorem says that the natural projection $TS^2 \mapsto S^2$ does not split.
On the other hand, for any nonempty topological spaces $X,Y$, the projection map $p_Y : X \times Y \to Y$ defined by $p_Y(x,y)=y$ always splits: pick $x_0 \in X$, and define $f(y)=(x_0,y)$.