The connection between domain in topology and Algebra

I first saw the word domain refer to connected open set in a complex analysis course.

Immediately, I observed the connection between algebra and geometry.

An open set $\mathrm{\Omega }$$\Omega$ is domain if and only if the holomorphic function ring over $\mathrm{\Omega }$$\Omega$ is an integral domain.

We know that a non-zero holomorphic function maps a domain to another domain.

Thus if you consider $f:{\mathrm{\Omega }}_1\to {\mathrm{\Omega }}_2$$f:\Omega_1\to\Omega_2$, then $f^{\ast }:H({\mathrm{\Omega }}_2)\to H({\mathrm{\Omega }}_1)$$f^*:H(\Omega_2)\to H(\Omega_1)$ will map an integral domain to another integral domain.

Thus holomorphic functions preserve domain in both algebra and geometry!

Riemann mapping theorem tells us that all the simply connected, proper open set of complex plane are conformal equivalence.

We might be able to prove it algebraically. Via this idea.

But for now, we can see that $H({\mathrm{\Omega }}_1)\cong H({\mathrm{\Omega }}_2)$$H(\Omega_1)\cong H(\Omega_2)$ if ${\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2$$\Omega_1,\Omega_2$ is simply connected, proper open set.