We give an overview of the basic definitions of condensed categories, as well as the internal Hom of condensed abelian groups. We give a construction for the internal Hom of condensed sets and apply it to obtain a new proof of a theorem of Clausen an
d Scholze. Finally, we give a detailed account of a construction of the real numbers from discrete spaces, which is an intermediate step of a theorem by Clausen and Scholze.
It is established that any homeomorphism between two closed negligible subset of $D^tau$ can be extended to an autohomeomorphism of $D^tau$.
Under a general categorical procedure for the extension of dual equivalences as presented in this papers predecessor, a new algebraically defined category is established that is dually equivalent to the category $bf LKHaus$ of locally compact Hausdor
ff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category $bf CLCA$ of complete local contact algebras and suitable morphisms. Thereby, a new proof is presented for the equivalence ${bf LKHaus}simeq{bf CLCA}^{rm op}$ that was obtained by the first author more than a decade ago. Unlike the morphisms of $bf CLCA$, the morphisms of the new category and their composition law are very natural and easy to handle.
We investigate the mathematics of a model of the human mind which has been proposed by the psychologist Jens Mammen. Mathematical realizations of this model consist of so-called emph{Mammen spaces}, where a Mammen space is a triple $(U,mathcal S,math
cal C)$, where $U$ is a non-empty set (the universe), $mathcal S$ is a perfect Hausdorff topology on $U$, and $mathcal Csubseteqmathcal P(U)$ together with $mathcal S$ satisfy certain axioms. We refute a conjecture put forward by J. Hoffmann-J{o}rgensen, who conjectured that the existence of a complete Mammen space implies the Axiom of Choice, by showing that in the first Cohen model, in which ZF holds but AC fails, there is a complete Mammen space. We obtain this by proving that in the first Cohen model, every perfect topology can be extended to a maximal perfect topology. On the other hand, we also show that if all sets are Lebesgue measurable, or all sets are Baire measurable, then there are no complete Mammen spaces with a countable universe. Finally, we investigate two new cardinal invariants $mathfrak u_M$ and $mathfrak u_T$ associated with complete Mammen spaces and maximal perfect topologies, and establish some basic inequalities that are provable in ZFC. We show $mathfrak u_M=mathfrak u_T=2^{aleph_0}$ follows from Martins Axiom, and, contrastingly, we show that $aleph_1=mathfrak u_M=mathfrak u_T<2^{aleph_0}=aleph_2$ in the Baumgartner-Laver model.
In this paper, we study some properties of the ring $C(X)_F$ of all real valued functions which are continuous except on some finite subsets of $X$. We show that $C(X)_F$ is closed under uniform limit if and only if the set of all non-isolated points
of $X$ is finite. We also initiate and investigate the zero divisor graph of the ring $C(X)_F$.
$C_p(X)$ denotes the space of continuous real-valued functions on a Tychonoff space $X$ endowed with the topology of pointwise convergence. A Banach space $E$ equipped with the weak topology is denoted by $E_{w}$. It is unknown whether $C_p(K)$ and $
C(L)_{w}$ can be homeomorphic for infinite compact spaces $K$ and $L$ cite{Krupski-1}, cite{Krupski-2}. In this paper we deal with a more general question: what are the Banach spaces $E$ which admit certain continuous surjective mappings $T: C_p(X) to E_{w}$ for an infinite Tychonoff space $X$? First, we prove that if $T$ is linear and sequentially continuous, then the Banach space $E$ must be finite-dimensional, thereby resolving an open problem posed in cite{Kakol-Leiderman}. Second, we show that if there exists a homeomorphism $T: C_p(X) to E_{w}$ for some infinite Tychonoff space $X$ and a Banach space $E$, then (a) $X$ is a countable union of compact sets $X_n, n in omega$, where at least one component $X_n$ is non-scattered; (b) $E$ necessarily contains an isomorphic copy of the Banach space $ell_{1}$.
We study new relations of the following statements with weak choice principles in ZF and ZFA. 1. For every infinite set X, there exists a permutation of X without fixed points. 2. There is no Hausdorff space X such that every infinite subset of X con
tains an infinite compact subset. 3. If a field has an algebraic closure then it is unique up to isomorphism. 4. Variants of Chain/Antichain principle. 5. Any infinite locally finite connected graph has a spanning subgraph omitting some complete bipartite graphs. 6. Any infinite locally finite connected graph has a spanning m bush for any even integer m greater than 4. We also study the new status of different weak choice principles in the finite partition model (a type of permutation model) introduced by Bruce in 2016. Further, we prove that Van Douwens Choice Principle holds in two recently constructed known permutation models.
In the present paper, we study the existence of best proximity pairs in ultrametric spaces. We show, under suitable assumptions, that the proximinal pair $(A,B)$ has a best proximity pair. As a consequence we generalize a well known best approximatio
n result and we derive some fixed point theorems. Moreover, we provide examples to illustrate the obtained results.
In this paper, we intend to show that under not too restrictive conditions, results much stronger than the one obtained earlier by Hejduk could be established in category bases.
We obtain several game characterizations of Baire 1 functions between Polish spaces X, Y which extends the recent result of V. Kiss. Then we propose similar characterizations for equi-Bare 1 families of functions. Also, using similar ideas, we give g
ame characterizations of Baire measurable and Lebesgue measurable functions.
