May 17th, 2017, 07:11 AM  #1 
Member Joined: Apr 2014 From: Greece Posts: 47 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: 2,487 Thanks: 630 
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, .

May 18th, 2017, 10:07 AM  #4 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,053 Thanks: 85 
An isomorphism is a bijective map which preserves structure.


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