Topological spaces in which countable intersections of dense open sets have dense interior

Question

In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense.

Now consider the following strengthening of the Baire property (let's temporarily call it “super Baire”): “a countable intersection of dense open sets has dense interior” (edit: to clear up a possible confusion, “dense” here means “dense in the whole space”; equivalently, “a countable union of nowhere dense sets is nowhere dense”, where “nowhere dense” means the closure has empty interior; reformulated: “a meager set is nowhere dense”).

An example of a topological space with this “super Baire” property is the real line with the density topology (because nowhere dense sets for the density topology are exactly the Lebesgue nullsets; see Tall, “The Density topology“, Pacific J. Math. 62 (1976) 275–284, theorem 2.7). Conversely, the real line with its usual (Euclidean) topology is Baire but not “super Baire” because the intersection of the open sets $\mathbb{R}\setminus\{r_n\}$ where $r_n$ ranges over all rationals, is the set $\mathbb{R}\setminus\mathbb{Q}$ of irrationals, which has empty interior.

Questions: Does this “super Baire” property have a standard name? Where might I learn more about it, or where might I find more examples of “super Baire” spaces?

Remarks (added in edit, 2024-08-04):

• Following a comment by “Not Mike”, I should have remarked that the “super Baire” notion I am asking about is reminiscent of the notion of “P-space”, which is a space in which a countable intersection of open sets is open (for facts about P-spaces, see this MO answer by Joseph Van Name to Countable intersections in topological space and the paper by Misra, “A Topological view of P-spaces”, General Topology App. 2 (1974) 349–362). But the notions are distinct: the density topology on $\mathbb{R}$ is “super Baire” but is not a P-space (because the intersection of the open — hence density-open — intervals from $-\frac{1}{n}$ to $\frac{1}{n}$ is $\{0\}$ which is not density-open); and I don't see a reason why a P-space should be super Baire (but neither do I see a counterexample).

• The “super Baire” property of $X$ seems equivalent to the assertion that the topos of sheaves on $X$ satisfies: $(\forall n:\mathbb{N}.\neg\neg P(n)) \Rightarrow (\neg\neg\forall n:\mathbb{N}.P(n))$ (“double-negation shift”) internally. Maybe this is relevant.

• Noé de Rancourt pointed out to me the following fact on Twitter: the set $[\mathbb{N}]^\infty$ of infinite subsets of $\mathbb{N}$ with the Ellentuck topology (i.e. the topology generated by the sets $[s, M] = \{N \;|\; s \sqsubseteq N ⊆ M\}$, where $s$ is finite, $M$ infinite, and $\sqsubseteq$ means “initial segment”). This is a consequence of Ellentuck's proof of Silver's theorem: every analytic subset of $[\mathbb{N}]^\infty$ is Ramsey. See for instance the book Introduction to Ramsey spaces by Stevo Todorcevic, section 1.4, for more detail: the “super Baire” property of the topology follows from lemma 1.53 (p. 23) there.

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Update (2024-08-05): The discussion has become a little bit confused because a number of similar statements have been mentioned so that we are (or at least, I am), lost in a maze of twisty little topological properties, all alike. Let me summarize:

1. The original Baire property: “a countable intersection of dense open sets is dense”.

2. The property I am asking about, and which I am provisionally calling “super-Baire”: “a countable intersection of dense open sets has a dense interior” (where “dense” means “dense in the whole space”).

3. The property mentioned under the name “property P” by YCor in a comment, but which is called “almost P-space” in the paper “P-spaces and the Volterra property” by Santi Spadaro: “a nonempty countable intersection of open sets has a nonempty interior” (equivalently: every if $E$ is a $G_\delta$ then the interior of $E$ is dense in $E$).

4. The property of being a P-space mentioned in the first of my remarks above: “a countable intersection of open sets is open” (i.e., every $G_\delta$ is open).

Obvious implications are: (2)⇒(1) and (4)⇒(3). Furthermore, (1)∧(3)⇒(2) (because if $E$ is a countable intersection of dense open sets, then (1) ensures that $E$ is dense in the whole space, and (3) ensures that the interior of $E$ is dense in $E$).

The usual topology on the real line satisfies (1) but neither (2) nor (3) nor (4). The density topology on the real line satisfies (1) and (2) but neither (3) nor (4). The Stone-Čech remainder (corona) $\mathbb{N}^* := \beta\mathbb{N}\setminus \mathbb{N}$ of a discrete countable set satisfies (1), (2) and (3) but not (4).

Answer 1

Let $X$ be a compact Hausdorff topological space put $A=C(X)$ the $C^*$ algebra of all complex valued continuous functions.

The Gelfand correspondence between the category of compact Hausdorff spaces and the category of commutative unital $C^*$ algebras leads us to a dictionary containing the following:(The Gelfand functor maps $X$ to $C(X)$)

An ideal correspond to an open set since for an open set $U$ one may consider the $I_U$ as the ideal of all continuous functions $f:X\to \mathbb{C}$ such that the support of $f$ is contained in $U$. An open dense set correspond to an essential ideal

Recall that an essential ideal in an algebra $A$ is an ideal $I$ with $I\cap J\neq \{0\}$ for every non zero ideal $J$.

The property you mentioned, a stronger version of the Baire property, is equivalent to the following:

Let $I_n$ be a sequence of essential ideals in $A=C(X)$ then $\cap_{n=1}^\infty I_n$ is an essential ideal in $A$

This property can be counted as non commutative analogy of the strong Baire property you indicated to

An advanced version of the dictionary I indicated to is mentioned here in Dictionary between commutative and non commutative geometry and topology

Update 2024-08-07 One application of the idea of investigation of the OP question via Gelfand correspondence between the category of commutative $C^*$ algebras and the category of compact Hausdorff spaces is the following:

As we see in the original question there is no any Hausdorff assumption. So one may pick a compact not necessarilly Hausdorff space $X$ which satisfies the super Baire property. Then check if the Hausdorfization $\tilde{X}$ of $X$ is again a super Baire space or not.

I explain about the Hausdorfization of a compact (not necessarily Hausdorff) space: Let $X $ be a compact topological space. Then $C(X)$ is a unital $C^*$ algebra. Then according to the Gelfand duality explained above there is a unique (up to homeomorphism) compact Hausdorff topological space $\tilde{X}$ such that $C(X)$ is $C^*$ isomorphic to $C(\tilde{X})$. The space $\tilde{X}$ obtained in this way is called the Hausdorfization of $X$.

It can be shown that the Hasdorfization $\tilde{X}$ of $X$ can be obtained as follows: We define an equivalent relation on $X$: $a\sim b$ if for every continuous function $f:X\to \mathbb{C}$ we necessarilly have $f(a)=f(b)$. The quotien space $\frac{X}{\sim}$ is the Hausdorfization of $X$ hence $\frac{X}{\sim}$ is a continuous open image of $X$. This implies that if $X$ is a super Baire space then its Hausdorfization is a super Baire space too. Because an open continuous image of a super Baire space is a Baire space.