Every Jordan curve inscribes uncountably many rhombi

Spaces of Uncountably Many Dimensions*

Riemann in his Habilitations Schrift of 1854 suggested the notion of ^-dimensional space (where n is a natural number) as an extension of the notion of three-dimensional euclidean space. Hubert extended the notion still further by defining a space of a countably infinite number of dimensions. Fréchetf in 1908 defined two other spaces of countably many dimensions, which he called D„ and J3W. Tyc...

UNCOUNTABLY MANY DIFFERENT INVOLUTIONS OF 5s

Extending Baire property by uncountably many sets

We prove that if ZFC is consistent so is ZFC + “for any sequence (An) of subsets of a Polish space 〈X, τ〉 there exists a separable metrizable topology τ ′ on X with B(X, τ) ⊆ B(X, τ ′), MGR(X, τ ′) ∩ B(X, τ) = MGR(X, τ) ∩B(X, τ) and An Borel in τ ′ for all n.” This is a category analogue of a theorem of Carlson on the possibility of extending Lebesgue measure to any countable collection of sets...

Uncountably Many Mildly Wild Non-wilder Arcs1

I. The basic example A0 (Figure 1). A regular normed projection of our basic example of a mildly wild non-Wilder arc is shown in Figure 1. (Using the methods of [4], one could easily give a precise description.) (A) Ao is not L.P.U. at p. The invariants of [7] will be used to show that the penetration index PiA0, p) of A0 at p is equal to 4. (For a definition of the penetration index see [l] an...

Digital Jordan Curve Theorems

Efim Khalimsky’s digital Jordan curve theorem states that the complement of a Jordan curve in the digital plane equipped with the Khalimsky topology has exactly two connectivity components. We present a new, short proof of this theorem using induction on the Euclidean length of the curve. We also prove that the theorem holds with another topology on the digital plane but then only for a restric...