Space Suggestion: Levy-Shapiro space

[Moniker1998]

Space Suggestion

Let $X_1$ be Fort Space on the Real Numbers and $X_2$ be Unit interval [0,1]. The space $X$ is the subspace of $\mathbb{R}\times [0, 1] \cup \{(\infty, 0)\}$ of $X_1\times X_2$.

Rationale

This space appears in the article Rings of quotients of rings of functions by Levy and Shapiro in example 3.7.

Relationship to other spaces and properties

This space provides an example of cozero complemented space such that there exists a cozero set whose closure is not a zero set. This shows that spaces with the latter property have the former, but not conversely.

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Does anyone know how spaces such that every cozero subset has zero set closure, are called, if anything?

[prabau]

Such spaces are called "basically disconnected". See the pending #1206.

Wait, maybe that's a different notion, right?

[Moniker1998]

@prabau such spaces are not called basically disconnected. That's a different property

[prabau]

Yes, it's different. Basically disconnected is a stronger property that implies the other one.

[Moniker1998]

@prabau Yes. Basically disconnected implies every closure of a cozero set is a zero set, which implies that the space is cozero complemented.

It'd be nice to add either the property of $F$-space or $F'$-space, since either together with cozero complemented are equivalent to basically disconnected. I think it would be nice to add $F$-spaces since they seem relatively popular property, and we have examples of such spaces on this site.

Then we can add two theorems, one that says basically disconnected spaces are $F$-spaces, and another which says that Tychonoff cozero complemented $F$-spaces are basically disconnected.