Two notes on formalized topology
Two problems on doubling measures.
Doubling measures appear in relation to quasiconformal mappings of the unit disk of the complex plane onto itself. Each such map determines a homeomorphism of the unit circle on itself, and the problem arises, which mappings f can occur as boundary mappings?
Two remarks concerning retracts of ANR-spaces
Two remarks on weaker connected topologies
It is shown that no generalized Luzin space condenses onto the unit interval and that the discrete sum of copies of the Cantor set consistently does not condense onto a connected compact space. This answers two questions from [2].
Two results concerning biconnected sets with dispersion points
Two Results on Jachymski-Schröder-Stein Contractions
We establish two fixed point theorems for certain mappings of contractive type.
Two results on special points
We show that there is a nowhere ccc σ-compact space which has a remote point. We show that it is consistent to have a non-compact σ-compact separable space X such that every point of the remainder is a limit of a countable discrete subset of non-isolated points of X. This example shows that one cannot prove in ZFC that every locally compact non-compact space has discrete weak P-points.
Two Selection Theorems
Two set-theoretic problems in topology
Two spaces homeomorphic to
We consider the spaces called , constructed on the set of all finite sequences of natural numbers using ultrafilters to define the topology. For such spaces, we discuss continuity, homogeneity, and rigidity. We prove that is homogeneous if and only if all the ultrafilters have the same Rudin-Keisler type. We proved that a space of Louveau, and in certain cases, a space of Sirota, are homeomorphic to (i.e., for all ). It follows that for a Ramsey ultrafilter , is a topological group....
Two theorems concerning BI-quotient maps
Two theorems concerning common fixed point of commutative mappings
Two theorems of the descriptive theory of point sets
Two theorems on the Scorza Dragoni property for multifunctions
We point out two theorems on the Scorza Dragoni property for multifunctions. As an application, in particular, we improve a Carathéodory selection theorem by A. Cellina [4], by removing a compactness assumption.
Two types of remainders of topological groups
We prove a Dichotomy Theorem: for each Hausdorff compactification of an arbitrary topological group , the remainder is either pseudocompact or Lindelöf. It follows that if a remainder of a topological group is paracompact or Dieudonne complete, then the remainder is Lindelöf, and the group is a paracompact -space. This answers a question in A.V. Arhangel’skii, Some connections between properties of topological groups and of their remainders, Moscow Univ. Math. Bull. 54:3 (1999), 1–6. It is...
Two-dimension-like functions defined on the class of all Tychonoff spaces.
Two-distance preserving functions.
Two-fold theorem on Fréchetness of products
A refined common generalization of known theorems (Arhangel’skii, Michael, Popov and Rančin) on the Fréchetness of products is proved. A new characterization, in terms of products, of strongly Fréchet topologies is provided.
Two-norm algebras
Two-sided nonsingular transformations