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I'm reading a First Course in Differential Geometry by Chuan-Chih Hsiung and on page 8 he says "A closed disk that is homeomorphic to $I^2$ [i.e. $I\times I$, where $I = [a, b]$] is connected. The surface $S^2$ of a 2-sphere can be expressed as the union of two closed disks with nonempty intersection."

I'm not sure what he means by the second sentence. Am I supposed to imagine two disks being deformed into the two halves of the sphere (so the disks touch each other at their circumferences)? I don't understand what it means to express the spherical surface as a union of two disks.

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Yes, that's exactly it. The northern and southern (closed) hemispheres of a sphere are topologically (closed) disks, intersecting on their boundaries. –  Robert Israel Nov 18 '12 at 6:47
    
And you can visualize their common boundary as the equator of the sphere. –  Brian M. Scott Nov 18 '12 at 7:28
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