how do I show that there is homeomorphism $f:H^2\longrightarrow S^2$, where $H^2$ is the closed upper hemisphere with antipodal equator points identified and $S^2$ is the sphere with antipodal points identified?
Well, note that a quotient space made from a compact space is compact. Consequently $H^2$ is compact. It is also Hausdorff. $S^2$ is Hausdorff, too. Any continuous bijection from a compact and Hausdorff space into a Hausdorff space is a homeomorphism. So it is enough to show that the natural bijection $f:H^2 \to S^2$ is continuous.
$f$ is the composition of the natural injection of a hemisphere (with quotient equator) into a sphere (with quotient equator) and the quotient map (from a sphere with quotient equator into a quotient sphere), so $f$ is continuous.
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