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This is exercise 1.3.8 in Hatcher:

Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of path connected, locally path-connected spaces $X$ and $Y$. Show that if $X\simeq Y$ then $\tilde{X}\simeq \tilde{Y}$.

I tried applying the lifting criterion, but I seem to be hitting a dead end. Any help would be much appreciated.

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    @Kara$t$ug: you might be interested in having a look at [this question](http://math.stackexchange.com/q/136405/5363)2012-04-24

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