Space Suggestion: (R × (0, +∞)) ∪ (Q × {0})

[yhx-12243]

I have not decided and found the name of this space yet. 😂

Space Suggestion

Let $X = \bigl( \mathbb R \times (0, + \infty) \bigr) \cup \bigl( \mathbb Q \times {0} \bigr)$ with subspace topology of $\mathbb R^2$.

Rationale

This space appears in https://math.stackexchange.com/questions/3003649, https://math.stackexchange.com/questions/3260430, and Wikipedia Baire_space.

Relationship to other spaces and properties

This space provides an example of a (nontrivial) Baire space that is not Strongly Choquet.

π-Base, Search for Baire + ~Empty + ~Strongly Choquet

[pzjp]

I would call it the rational edge halfplane.
Or a fringed carpet space if aiming for something whimsy.