Let $(X, d)$ be a compact metric space and $f:X\to X$ be a homeomorphism such that $f(a)= a$ and $X\cap \mathbb{N}=\emptyset$.
In my research I need to construct a compact pseudo metric $(X\cup\mathbb{N}, \rho)$ and homeomorphism $g$ on $X\cup\mathbb{N}$ such that $g|X=f$ and $g(n)=n$ for all $n\in\mathbb{N}$. For this purpose in the following, I give a solution for it:
Let $Y=X\cup\mathbb{N}$ and $\rho:Y\times Y\to[0, \infty)$ define by $\rho(n, m)=0$ and $\rho(x, n)= d(x, a)$ and $\rho(m, y)=d(a, y)$ if $x,y\in X$ and $n,m\in\mathbb{N}$. Also $g:Y\to Y$ be defined by $g(y)= f(y)$ if $y\in X$ and $g(n)=n$ for $n\in\mathbb{N}$.
Please help me to know that my proof is true?