Where can I read about the topological properties of the perforated plane? Anybody knows about perforated plane in topology? What is it? Where can I read about it? I'm talking about the plane $\mathbb{R}^2$ with the topology that have basis elements disks without finite lines passing through the center of the disks. Not $\mathbb{R}^2- \\{0\\}$. I did not find nothing about it in the web, nor in wikipedia. Perhaps this is not the correct name, the name "plano perforado" is in spanish. Thanks.

I'm not at all sure, but I _think_ you might be describing what Seebach and Steen ( _Counterexamples in Topology_ ) refer to as the “deleted diameter topology”. Their description is:

> Let $X$ be the Euclidean plane, we define the deleted diameter topology on $X$ by taking as a subbasis for a topology $\sigma$ all open discs with all of the horizontal diameters other than the center, excluded.

(Section 76, page 95.)

That is, the sub-basic open sets consist of sets of the form $(C\setminus D_C)\cup\\{O_C\\}$ where $C$ is a disc, $D_C$ the diameter of the disc, and $O_C$ the center of the disc.

Google search for “deleted diameter topology” produces a number of hits, including an item on the $\pi$-base web site. Here's the page about it on Austin Mohr's “Spacebook” web site.