I've just begun reading Spivak's Calculus on Manifold and attempted to proof this simple result.
-I've updated my proof-
My proof are as follows,
img http://dl.dropbox.com/u/5681270/open-set%20proofs.png
My proof for the intersection case still looks kinda dubious though.
@Devan Ware, the notation $N_{\epsilon}(x)$ looks very useful to me but i haven't seen it anywhere, which branch of math is it found in and where can i learn more about it?