ℒ denotes the Lebesgue measurable subsets of ℝ and ${ℒ}_0$ denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ0$hasaperfectsubsetQ\in ℒ\${ℒ}_0$ which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes ${ℒ}_0$). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal $\left(s^0\right)$ which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has...

We prove that, assuming MA, every crowded $T_0$ space $X$ is $\omega$-resolvable if it satisfies one of the following properties: (1) it contains a $\pi$-network of cardinality $<𝔠$ constituted by infinite sets, (2) $\chi \left(X\right)<𝔠$, (3) $X$ is a $T_2$ Baire space and $c\left(X\right)\le {\aleph }_0$ and (4) $X$ is a $T_1$ Baire space and has a network $𝒩$ with cardinality $<𝔠$ and such that the collection of the finite elements in it constitutes a $\sigma$-locally finite family. Furthermore, we prove that the existence of a $T_1$ Baire irresolvable space is equivalent to the existence of...

We show that the existence of a non-trivial category base on a set of regular cardinality with each subset being Baire is equiconsistent to the existence of a measurable cardinal.

Let C denote the Banach space of real-valued continuous functions on [0,1]. Let Φ: C × C → C. If Φ ∈ +, min, max then Φ is an open mapping but the multiplication Φ = · is not open. For an open ball B(f,r) in C let B²(f,r) = B(f,r)·B(f,r). Then f² ∈ Int B²(f,r) for all r > 0 if and only if either f ≥ 0 on [0,1] or f ≤ 0 on [0,1]. Another result states that Int(B₁·B₂) ≠ ∅ for any two balls B₁ and B₂ in C. We also prove that if Φ ∈ +,·,min,max, then the set ${\Phi }^{-1}\left(E\right)$ is residual whenever E is residual in...