Where can I read about the topological properties of the perforated plane?

Question

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.

Answer 1

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.