General Topology
\begin{abstrac} Let $(X,T) $ be a topological space, and $^{*}X$ a non--standard extension of $X$. There is a natural ``standard'' topology $^{S}T$ on $^{*}X$ generated by $^{*}G$, where $G\in T$. The topological space $(^{*}X,^{S}T) $ will…
We prove that a transfinite extension of asymptotic dimension asind is trivial. We introduce a transfinite extension of asymptotic dimension asdim and give an example of metric proper space which has transfinite infinite dimension.
In this work, topological spaces are enriched by additional structures in order to give a more realistic representation of real life phenomena and computational processes and at the same time, to provide for utilization of the powerful…
We consider the Complex Stone-Weierstrass Property (CSWP), which is the complex version of the Stone-Weierstrass Theorem. If X is a compact subspace of a product of three linearly ordered spaces, then X has the CSWP if and only if X has no…
We investigate some topological properties of a normal functor $H$ introduced earlier by Radul which is some functorial compactification of the Hartman--Mycielski construction HM. We prove that the pair ($HX$, HM$Y$) is homeomorphic to the…
We show that there exists a natural counterpart of the Gromov-Hausdorff metric in the class of ultrametric spaces. It is proved, in particular, that the space of all ultrametric spaces whose metric take values in a fixed countable set is…
In this paper we settle all questions whether (it is consistent that) the properties P and Q [do not] coincide, where P and Q run over selection principles of the type U_fin(O,A).
In this paper we formulate three problems concerning topological properties of sets generating Borel non-sigma-compact groups. In case of the concrete F_\sigma\delta-subgroup of the Cantor group this gives an equivalent reformulation of the…
In this paper we characterize [strict] o-boundedness of the free (abelian) topological group F(X) (A(X)) as well as the free locally-convex linear topological space L(X) in terms of properties of a Tychonoff space X. These properties appear…
This paper deals mainly with the Chu duality of discrete groups. Among other results, we give sufficient conditions for an $FC$ group to satisfy Chu duality and characterize when the Chu quasi-dual and the Takahashi quasi-dual of a group…
We show there is no categorical metric continuum. This means that for every metric continuum X there is another metric continuum Y such that X and Y have (countable) elementarily equivalent bases but X and Y are not homeomorphic. As an…
We prove addition and subspace theorems for asymptotic large inductive dimension. We investigate a transfinite extension of this dimension and show that it is trivial.
In this paper, we show that a pointwise-symmetric isotonic closure function is uniquely determined by the pairs of sets it separates. We then show that when the closure function of the domain is isotonic and the closure function of the…
We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an…
If B is a compact space and B\{pt} is Lindelof then B^k\{pt} is star-Linedlof for every cardinality k. If B\{pt} is compact then B^k\{pt} is discretely star-Lindelof. In particular, this gives new examples of Tychonoff discretely…
Assuming Jensen's principle diamond, there is a compact Hausdorff space X which is hereditarily Lindelof, hereditarily separable, and connected, such that no closed subspace of X is both perfect and totally disconnected. The Proper Forcing…
Consider the self-map F of the space of real-valued test functions on the line which takes a test function f to the test function sending a real number x to f(f(x))-f(0). We show that F is discontinuous, although its restriction to the…
We proof a foliated version of the Poincare-Hopf theorem and other results which clarify the geometric and ergodic meaning of the Euler characteristic of a measured foliation.
We introduce notions of nearly good relations and N-sticky modulo a relation as tools for proving that spaces are D-spaces. As a corollary to general results about such relations, we show that C_p(X) is hereditarily a D-space whenever X is…
In this paper the notion of $\infty$-open-multicommutativity of functors in the category of compact Hausdorff spaces is considered. This property is a generalization of the open-multicommutativity on the case of infinite diagrams. It is…