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 May 17th, 2017, 07:11 AM #1 Member   Joined: Apr 2014 From: Greece Posts: 58 Thanks: 0 Isomorphic spaces I stumbled upon this symbol ≅ and I found out it stands for isomorphism. Can anybody explain what it means for a space D to be isomorphic to R (D ≅ R) ?
 May 17th, 2017, 08:34 AM #2 Member     Joined: Oct 2014 From: Colorado Posts: 40 Thanks: 21 Two spaces $A$ and $B$ are isomorphic if there exists a bijective map $\phi : A \rightarrow B$. If two spaces are isomorphic then it means they are similar but labeled differently. Check this out https://en.wikibooks.org/wiki/Linear...f_Isomorphisms
 May 18th, 2017, 03:57 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,189 Thanks: 871 What kind of "spaces" are you talking about? If they are "vector spaces" (or "linear spaces) in linear algebra, then that bijection must also "preserve the operations". That is, $\phi(au+ bv)= a\phi(u)+ b\phi(v)$.
 May 18th, 2017, 10:07 AM #4 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,364 Thanks: 100 An isomorphism is a bijective map which preserves structure.

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