As the title says, I'm trying to fill in the blank spots Conway leaves to show that if X is Tychonoff and $\Omega$ is a compact space satisfying:
a) $\pi:X\rightarrow\Omega$ is a homeomorphism onto its image
b) $\pi(X)$ is dense in $\Omega$
c) For every$f\in C_b(X)$, there exists $f'\in C(\Omega)$ such that $f=f'\circ\pi$
Then there exists a homeomorphism $g:\beta X\rightarrow \Omega$.
For background, Conway constructs $\beta X$ by considering it as a subset of $C_b(X)^*$, namely, the closure of the image of the map $\Delta:X\rightarrow (C_b(X)^*,wk^*):x\mapsto \delta_x$
My problem is the following:
Let $\{x_i\}$ be a net in X such that $\Delta(x_i)\rightarrow\tau_0\in \beta X$. Then $\{\pi(x_i)\}$ is a net in $\Omega$ and since $\Omega$ is compact, there exists a $\omega_0 \in \Omega$ such that it is a cluster point of $\pi(x_i)$. Using the properties of $\beta X$, It is pretty easy to show that $\omega_0$ is in fact the unique cluster point of $\{\pi(x_i)\}$ and so $\{\pi(x_i)\}\rightarrow \omega_0$. I want to show that if $\{y_i\}$ is another net in $X$ such that $\Delta(y_i)\rightarrow \tau_0$, then we must have that $\{\pi(y_i)\}\rightarrow \omega_0$. I'm able to show that if $\{\pi(y_i)\}\rightarrow \omega_0'$, then no element of $C(\Omega)$ can distinguish between $\omega$ and $\omega’$. I don't know where to go from here, or if this is even the right direction to be going in.