Weak continuity of fuzzy-valued maps
Weak continuity properties of topologized groups
We explore (weak) continuity properties of group operations. For this purpose, the Novak number and developability number are applied. It is shown that if is a regular right (left) semitopological group with such that all left (right) translations are feebly continuous, then is a topological group. This extends several results in literature.
Weak contractibility and hyperspaces
Weak covering properties and the class MOBI
Weak extent in normal spaces
If is a space, then the weak extent of is the cardinal If is an open cover of , then there exists such that and . In this note, we show that if is a normal space such that and , then does not have a closed discrete subset of cardinality . We show that this result cannot be strengthened in ZFC to get that the extent of is smaller than , even if the condition that is replaced by the stronger condition that is separable.
Weak forms of continuity and associated properties.
Weak forms of continuity in fuzzy topology
Weak homotopy equivalences of mapping spaces and Vogt's lemma
Weak L-structures and dimension
Weak openness and almost openness.
Weak orderability of second countable spaces
We demonstrate that a second countable space is weakly orderable if and only if it has a continuous weak selection. This provides a partial positive answer to a question of van Mill and Wattel.
Weak orderability of some spaces which admit a weak selection
We show that if a Hausdorff topological space satisfies one of the following properties: a) has a countable, discrete dense subset and is hereditarily collectionwise Hausdorff; b) has a discrete dense subset and admits a countable base; then the existence of a (continuous) weak selection on implies weak orderability. As a special case of either item a) or b), we obtain the result for every separable metrizable space with a discrete dense subset.
Weak selections and flows in networks
We demonstrate that every Vietoris continuous selection for the hyperspace of at most 3-point subsets implies the existence of a continuous selection for the hyperspace of at most 4-point subsets. However, in general, we do not know if such ``extensions'' are possible for hyperspaces of sets of other cardinalities. In particular, we do not know if the hyperspace of at most 3-point subsets has a continuous selection provided the hyperspace of at most 2-point subsets has a continuous selection.
Weak selections and weak orderability of function spaces
It is proved that for a zero-dimensional space , the function space has a Vietoris continuous selection for its hyperspace of at most 2-point sets if and only if is separable. This provides the complete affirmative solution to a question posed by Tamariz-Mascarúa. It is also obtained that for a strongly zero-dimensional metrizable space , the function space is weakly orderable if and only if its hyperspace of at most 2-point sets has a Vietoris continuous selection. This provides a partial...
Weak-bases and -spaces
It is shown that certain weak-base structures on a topological space give a -space. This solves the question by A.V. Arhangel’skii of when quotient images of metric spaces are -spaces. A related result about symmetrizable spaces also answers a question of Arhangel’skii. Theorem.Any symmetrizable space is a -space hereditarily. Hence, quotient mappings, with compact fibers, from metric spaces have a -space image. What about quotient -mappings? Arhangel’skii and Buzyakova have shown that...
Weak-chainability of tree-like continua and the combinatorial properties of mappings
Weak-continuity and closed graphs
Weakly almost periodic mappings on one- and two-manifolds
Weakly Borel-complete topological spaces
Weakly chainable circle-like continua