How to visualize the one point compactification of a plane with finitely points removed and $\mathbb{R}_{\text{discrete}}$ I wish to visualize the one point compactification of $\mathbb{R}^2$ with finitely points removed and that of $\mathbb{R}_{\text{discrete}}$
For the first question, I can picture it with one point removed. The resulting shape is a hollow sphere containing the removed point. The sphere is closed and bounded set of $\mathbb{R}^3$ so it is compact. But I cannot for the life of me picture what would happen if finite points were removed. Is it some sort of nested sphere?
I am also not sure what would the one point compactification of $\mathbb{R}_{\text{discrete}}$ be. It seems to be clear that for the usual topology, we simply wrap it to a circle. But what about the discrete topology?
Thanks for your help!!

Judging from your description, you’re probably not visualizing the one-point compactification of $\Bbb R^2\setminus\{p\}$ correctly. The one-point compactification of $\Bbb R^2\setminus\{p\}$ can be visualized as a horn torus, a torus with inner radius $0$. It’s what you get if you start with $S^2$ and identify the north and south poles. If $F\subseteq\Bbb R^2$ is finite, the one-point compactification of $\Bbb R^2\setminus F$, is what you get if you start with a sphere $S^2$ and identify $|F|+1$ points.
As for $\Bbb R_{\text{discrete}}$, Remember that the open nbhds of the new point are the complements of the compact sets in $\Bbb R_{\text{discrete}}$. The compact sets in $\Bbb R_{\text{discrete}}$ are precisely the finite sets, so the open nbhds of the new point are the cofinite sets containing it. If $p$ is the new point, the topology is
$$\big\{\{p\}\cup(\Bbb R\setminus F):F\text{ is finite}\big\}\cup\wp(\Bbb R)\;.$$
