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 ?

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- May 15th 2012, 09:58 PMmonsterhomeomorphism 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 PMDevenoRe: 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 AMmonsterRe: homeomorphism help
Is it that it cant because a homeomorphic function is bijective and continuous?

- May 16th 2012, 01:05 AMSylvia104Re: Homeomorphism help
Look closely at the definition of pre-image: If $\displaystyle f:A\to B$ and $\displaystyle V\subseteq B,$ the pre-image of $\displaystyle V$ under $\displaystyle f$ is $\displaystyle f^{-1}(V)=\{x\in A:f(x)\in V\}.$ In other words, $\displaystyle x\in f^{-1}(V) \Leftrightarrow f(x)\in V$ for all $\displaystyle x\in A.$ Now try assuming $\displaystyle f^{-1}(\O)\ne\O$ and see what happens.

- May 16th 2012, 01:37 AMmonsterRe: 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 AMSylvia104Re: Homeomorphism help