Related papers: Uncountable n-Dimensional Excellent Regular Local …

We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.

For any particularly interesting theorem one proof is never enough. Instead, the first proof sets the challenge to find a more elegant method that illuminates subtle features of the math, is simpler to understand, or even avoids using…

We give a short proof of the well-known fact that the unit interval [0,1] is uncountable by means of a simple infinite game. We also show using this game that a (non-empty) perfect subset of [0,1] must be uncountable.

We show that a suitable ring with a ``nice'' topology, in which convergent limits of units are units, is an \aleph_0-exchange ring. We generalize the argument to show that a semi-regular ring, R, with a ``nice'' topology, is a full exchange…

The main result of this article is: THEOREM. Every homogeneous locally conical connected separable metric space that is not a $1$-manifold is strongly $n$-homogeneous for each $n \geq 2$ and countable dense homogeneous. Furthermore,…

We show a number of undecidable assertions concerning countably compact spaces hold under PFA(S)[S]. We also show the consistency without large cardinals of "every locally compact, perfectly normal space is paracompact".

We introduce the notion of indivisible sequences and show that to any indivisible sequence $\{S, \Psi: S \to R\}$ we can associate faithfully flat ring maps $R \to R'$ that are not descendable. As a corollary, we obtain the first example of…

Let $n$ be a positive integer. An $n$-cycle of linear mappings is an $n$-tuple $(u_1,\dots,u_n)$ of linear maps $u_1 \in \mathrm{Hom}(U_1,U_2),u_2 \in \mathrm{Hom}(U_2,U_3),\dots,u_n \in \mathrm{Hom}(U_n,U_1)$, where $U_1,\dots,U_n$ are…

The $\pi$-weight spectrum for countable regular Fr\'echet-Urysohn spaces is the set of uncountable cardinals that are equal to the $\pi$-weight for some such space. We determine this $\pi$-weight spectrum in the standard Miller rational…

We prove that there exists a countable infinite sequence of non-empty special $\Pi^0_1$ classes $\{\mathcal{P}_i\}_{i\in\omega}$ such that no infinite union of elements of any $\mathcal{P}_i$ computes the halting set. We then give a…

We compare two examples of random dense countable sets, `Brownian local minima' and `unordered uniform infinite sample'. They appear to be identically distributed. A framework for such notions is proposed. In addition, random elements of…

A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed; the former ad hoc proof of this…

Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. We prove that such a set exists if $X$ is…

There has arisen in recent years a substantial theory of "multiplier ideals'' in commutative rings. These are integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among…

Given a coloring of the k-element subsets of an uncountable separable metric space, we show that there exists an infinite monochromatic subset which contains its limit point.

We show every locally solvable subgroup of PLo(I) is countable. A corollary is that an uncountable wreath product of copies of $\mathbb{Z}$ with itself does not embed into PLo(I).

Cantor's famous proof of the non-denumerability of real numbers does apply to any infinite set. The set of exclusively all natural numbers does not exist. This shows that the concept of countability is not well defined. There remains no…

Let $X$ be a topological vector space of complex-valued sequences and $Y$ be a subset of $X$. We provide conditions for $X \setminus Y \cup \{0\}$ to contain uncountably infinitely many linearly independent dense vector subspaces of $X$. We…

A result of Watanabe and Yoshida says that an unmixed local ring of positive characteristic is regular if and only if its Hilbert-Kunz multiplicity is one. We show that, for fixed $p$ (characteristic) and $d$ (dimension), there exist a…

We make use of a finite support product of Jensen forcing to define a model in which there is a countable non-empty lightface $\Pi^1_2$ set of reals containing no ordinal-definable real.