Space Suggestion: Cross topology on $\mathbb{R}^2$

[Moniker1998]

Space Suggestion

Let $X = \mathbb{R}^2$ and call $U$ open if for all $(x, y)\in U$ there is a cross $C((x, y), r)\subseteq U$ where $$C((x, y), r) = \{x\}\times (-r+y, r+y)\cup (-r+x, r+x)\times\{y\}$$

Rationale

This topology appears in Topological groups and related structures by Arhangelskii and Tkachenko, example 1.2.6

I also came up with it while considering topologies on posets. Namely if we apply my construction from this answer to the $\mathbb{R}^2$ with order $(x, y) < (z, t)$ iff $x < z$ and $y < t$ (which is, as far as I can tell, the product in the category of posets), then we obtain cross topology.

Relationship to other spaces and properties

Cross topology is stronger than radial plane, which is stronger than the Euclidean plane.
I believe it has different properties from the radial plane, something that we might be able to show by adding this space to pi-base and comparing the properties of both.