Open problems on countable dense homogeneity
... We do not know to whom this problem belongs. By the result of Kennedy [48], for continua this is equivalent to asking whether every 2-homogeneous metric continuum admits a homeomorphism that is the identity on some non-empty open set. On the other hand, there are even Baire SLH connected separable m ...
... We do not know to whom this problem belongs. By the result of Kennedy [48], for continua this is equivalent to asking whether every 2-homogeneous metric continuum admits a homeomorphism that is the identity on some non-empty open set. On the other hand, there are even Baire SLH connected separable m ...
Categorically proper homomorphisms of topological groups
... and compare it with the weaker notions of c-completeness and h(omomorphical) completeness as introduced here at the morphism level as well, in generalization of the object notion of h-completeness studied in [12, 19]. To derive product stability of c-proper and h-complete maps, we extend the charact ...
... and compare it with the weaker notions of c-completeness and h(omomorphical) completeness as introduced here at the morphism level as well, in generalization of the object notion of h-completeness studied in [12, 19]. To derive product stability of c-proper and h-complete maps, we extend the charact ...
Lecture 5
... One disadvantage of the Priestley duality is that it requires both topology and order to represent distributive lattices. We will show that we can do away with the order. To see this, let L be a bounded distributive lattice. We again work with the set X (L) of prime filters of L. But now we ignore s ...
... One disadvantage of the Priestley duality is that it requires both topology and order to represent distributive lattices. We will show that we can do away with the order. To see this, let L be a bounded distributive lattice. We again work with the set X (L) of prime filters of L. But now we ignore s ...
Almost disjoint families and topology
... Almost disjoint families of graphs of functions were studied in various contexts [37, 99, 133, 165] though many fundamental problems remain open. An important contribution to the subject has recently been made by Raghavan [134] who showed that in ZFC there is an AD family of graphs of functions whic ...
... Almost disjoint families of graphs of functions were studied in various contexts [37, 99, 133, 165] though many fundamental problems remain open. An important contribution to the subject has recently been made by Raghavan [134] who showed that in ZFC there is an AD family of graphs of functions whic ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.