homeomorphism help

• May 15th 2012, 09:58 PM
monster
homeomorphism help
Hi, learning about homemorphic functions at the moment and am failry confused,

If i have a homeomorphic function h: A - B between two topological spaces A and B,

Is the pre image of the empty set in B the empty set in A ?
• May 15th 2012, 10:06 PM
Deveno
Re: homeomorphism help
let me ask you this: can the pre-image of the empty set in B have ANY elements in it?
• May 16th 2012, 12:05 AM
monster
Re: homeomorphism help
Is it that it cant because a homeomorphic function is bijective and continuous?
• May 16th 2012, 01:05 AM
Sylvia104
Re: Homeomorphism help
Look closely at the definition of pre-image: If $f:A\to B$ and $V\subseteq B,$ the pre-image of $V$ under $f$ is $f^{-1}(V)=\{x\in A:f(x)\in V\}.$ In other words, $x\in f^{-1}(V) \Leftrightarrow f(x)\in V$ for all $x\in A.$ Now try assuming $f^{-1}(\O)\ne\O$ and see what happens.
• May 16th 2012, 01:37 AM
monster
Re: Homeomorphism help
i'm sorry i understand the definition but i think i may have some understanding of the empty set missing, obviously this cant happen but why cant a function map something to the empty set and the empty set to something else
• May 16th 2012, 03:17 AM
Sylvia104
Re: Homeomorphism help
Quote:

Originally Posted by monster
why cant a function map something to the empty set and the empty set to something else

We're not talking about the empty set as an element of $A$ or $B;$ we're talking about it as a subset of $A$ and $B.$