I am attempting to solve Exercise 4.4.7 from this note:
Given a variety $X$ and a closed subset $Y \subseteq X$ equipped with the induced topology. For $V \subset Y$ open define $$\mathcal{O}_Y(V)=\{f: V \to k \mid \forall P \in V, \exists \text{ an open neighborhood } U \subseteq X \text{ containing } P,\exists g \in \mathcal{O}_X(U) \text{ such that } f=g \text{ on } V \cap U\}.$$
I have proved successfully that $(Y,\mathcal{O}_Y)$ is a $k$-space, however I am stuck at the step proving it is an algebraic variety. My attempt: For every $x \in X$, call $U_x$ an open subset of $X$ containing $x$. Then we can split $(X,\mathcal{O}_X)$ into smaller spaces $(U_x,\mathcal{O}_{X \mid U_x})$, each of which is isomorphic to $(Y_x,\mathcal{O}_{Y_x})$ where $Y_x$ is closed in some $\mathbb{A}^n$ and $\mathcal{O}_{Y_x}$ is the sheaf of regular function. Note that $Y=\cup_{x \in X}(Y \cap U_x)$. Call $$\varphi_x: (U_x,\mathcal{O}_{X \mid U_x}) \xrightarrow[]{\sim} (Y_x,\mathcal{O}_{Y_x}).$$ Since $U_x \cap Y$ is open in $U_x$, $\varphi_x(U_x \cap Y)$ is open in $Y$. Now we define $$\widetilde{\varphi}_x=\varphi_{x \mid U_x \cap Y}: (U_x \cap Y,\mathcal{O}_{Y \mid U_x \cap Y}) \rightarrow (\varphi_x(U_x \cap Y),\mathcal{O}_{Y_x \mid \varphi(U_x \cap Y)}).$$ I intend to show that $\widetilde{\varphi}_x$ is indeed an isomorphism of $k$-spaces by showing that regularity at $\widetilde{\varphi}_x(P)$ is equivalent to regularity at $P$, and also show the RHS is isomorphic to some algebraic variety. I got stuck here.
I have read many other relevant posts but most of them don't express explicitly the proof that the RHS is a variety. That's why I write this post with my own attempt. However if I was dumb at any argument, I am really happy to be pointed out those mistakes and hints to fix them.
Any helps or hints are really appreciated.