Let $M$, $N$, $A$, $B$ be topological spaces (or manifold) such that $A$ and $B$ are subspaces in $M$, $N$ respectively.
Let $f: A \to B$ and $g:A \to B$ be homeomorphism and assume that $f$ and $g$ are isotopic. We attach spaces $M$ and $N$ via $f$ and $g$ and obtain $M\cup_fN$ and $M\cup_gN$.
I want to prove (or disprove) that $M\cup_fN$ and $M\cup_gN$ are homeomorphic.
First of all, I am confused by the difinition of isotopy. My understanding is that homeomorphisms $f: A \to B$ and $g:A \to B$ are isotopic if there is a map $H: A\times [0,1] \to B$ such that $H(x, t)$ is a homeomorphism for each $t\in [0,1]$ and $H(x, 0)=f(x)$ and $H(x,1)=g(x)$. Is this definition correct in this context?
Any suggestions to the definition of isotpy and construction of a homeomorphism between $M\cup_fN$ and $M\cup_gN$ are appreciated.