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Motivation: I understand that various attempts have been made at defining a topology on $\mathbb{R}$ that is an analogue of the density topology ([1]) but for category (and meager sets) instead of measure (and null sets): see, e.g., [2] and [3]. However, these attempts focus on the notion of a density point and the topologies they define are not T3½ (see [2], theorem 5). I wonder if a different kind of analogue can be found whose topological and descriptive set-theoretical properties are closer to the density topology at the price, perhaps, of a more distant connection with the notion of “density point”.

Question: Is there a topology $\mathscr{T}$ on $\mathbb{R}$ which is:

  • Hausdorff and completely regular (T3½),

  • connected,

  • translation-invariant (in the sense that the $x\mapsto x+y$ are homeomorphisms),

  • finer than the usual (Euclidean) topology on $\mathbb{R}$,

  • such that the regular-open sets for $\mathscr{T}$ (= sets equal to the interior of their closure) are Borel for the usual topology, and

  • such that the nowhere-dense sets for $\mathscr{T}$ (= sets whose closure has empty interior) are closed for $\mathscr{T}$ and are exactly the meager sets for the usual topology.

(These properties are all satisfied by the density topology except that in the last one, “meager” should be replaced by “Lebesgue measure zero”: see [1], §1. I don't know if they characterize it, but I have no reason to think so.)

Note: I am more interested in a positive answer than a negative one, so the properties listed above are somewhat negotiable (in the sense that I am also interested in positive answers satisfying a reasonable subset of the above): most important are the T3½ property and the fact that the Boolean algebra of regular-open sets for $\mathscr{T}$ should be the Boolean algebra of Borel sets modulo meager sets for the usual topology.

References:

  1. F. Tall, “The Density Topology”, Pacific J. Math. 62 (1976) 275–284.

  2. W. Poreda, E. Wagner-Bojakowska & W. Wilczyński, “A category analogue of the density topology”, Fundam. Math. 125 (1985) 167–173.

  3. W. Wilczyński, “A category analogue of the density topology non-homeomorphic with the $\mathcal{I}$-density topology”, Positivity (2018).

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