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

[yhx-12243]

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

Space Suggestion

Let X = (ℝ × (0, +∞)) ∪ (ℚ × {0}) with subspace topology of ℝ².

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.

[prabau]

Other possible name: half-plane with rationals on the boundary ?

[prabau]

As an aside, the space S75 (Rational tangent disk topology) has the same underlying set and a finer topology.

Personally I don't like the name "Rational tangent disk topology". Not sure how it has been named in the literature, if at all; but I think it would be good to find a better name.