While the result of Young mentioned in the answer of Greg Kuperberg seems to be of purely topological interest, quite surprisingly, it has applications to study of the **structure of the support of Sobolev functions**: 
T. Bagby, P. M. Gauthier,
<A HREF="https://cms.math.ca/10.4153/CMB-1998-037-4"><FONT FACE="Arial">Note on the support of Sobolev functions</FONT></A><FONT FACE="Arial">.
*Canad. Math. Bull.* 41 (1998), no. 3, 257–260. 
(<A HREF="https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=RT&pg7=ALLF&pg8=ET&review_format=html&s4=bagby%20and%20g%2A&s5=&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=1637637"><FONT FACE="Arial">MathSciNet  review</FONT></A><FONT FACE="Arial">). The cited paper contains a proof (Lemma 2 in the paper) since the authors knew the result of Moore, but they were not aware of the generalization obtained by Young.