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In Aguilar & Guitler and Prieto page 63, I read the following proposition:

Let $X = X_1\cup X_2$ with $X_1$, $X_2$ open subsets. If $X_1$ and $X_2$ are simply connected and $X_1\cap X_2$ is path connected, then $X$ is simply connected.

Next, in order to show that the sphere $S^n$ is simply connected they use this proposition with $X_1=S^n-N$ and $X_2=S^n-S$ where $N$ and $S$ are the north and south poles. My try was to choose $X_1=H_+$ and $X_2=H_-$ where $H_+$ and $H_-$ are the upper and lower hemispheres which are homeomorphic to discs so they are contractible (in particular simply connected) and their intersection is the sphere $S^{n-1}$ which is path connected hence by the previous proposition, $S^n$ is simply connected. Is my try correct ? thank you for your help!

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    You need to be a bit careful because the proposition requires open subsets, and what you've described are not. The hemispheres should stretch a bit further past the equator, so that their intersection is a small neighbourhood around the equator. This would then be equivalent to what the authors wrote. – 2017-02-08

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