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If $q: E\rightarrow X$ is a covering map that has a section (i.e. $f: X\rightarrow E, q\circ f=Id_X$) does that imply that $E$ is a $1$-fold cover?

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    Does it have *only one* section?2012-12-12
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    Well, what if $E = X \amalg X$?2012-12-12
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    @ZhenLin: I forgot to add that $E$ has to be connected...because that obviously would not hold in case $E$ is not connected, as you pointed out.2012-12-12
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    @Andy I'm not sure how that makes a difference?2012-12-12
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    Well, think about $\mathbb{R}$ covering $S^1$, or $\mathbb{C}\setminus \{ 0 \}$ covering itself with the map $z \mapsto z^n$.2012-12-12
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    Addressing the other question by OP in the answers, I would like to suggest the following: $f\colon x \mapsto (x,0)$ and $q\colon(x,y)\mapsto x$2012-12-12
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    For $p$ to have a section, does $p$ have to be a homeomorphism?2012-12-12

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