Shape properties of hyperspaces
Shape theory intrinsically.
We prove in this paper that the category HM whose objects are topological spaces and whose morphisms are homotopy classes of multi-nets is naturally equivalent to the shape theory Sh. The description of the category HM was given earlier in the article "Shape via multi-nets". We have shown there that HM is naturally equivalent to Sh only on a rather restricted class of spaces. This class includes all compact metric spaces where a similar intrinsic description of the shape category using multi-valued...
Shape theory of maps.
We shall describe a modification of homotopy theory of maps which we call shape theory of maps. This is accomplished by constructing the shape category of maps HMb. The category HMb is built using multi-valued functions. Its objects are maps of topological spaces while its morphisms are homotopy classes of collections of pairs of multi-valued functions which we call multi-binets. Various authors have previously given other descriptions of shape categories of maps. Our description is intrinsic in...
Shapes of finite-dimensional compacta
Sharkovskiĭ's theorem holds for some discontinuous functions
We show that the Sharkovskiĭ ordering of periods of a continuous real function is also valid for every function with connected graph. In particular, it is valid for every DB₁ function and therefore for every derivative. As a tool we apply an Itinerary Lemma for functions with connected graph.
Sheaves and -bicompactifications of mappings.
Sierpiński's hierarchy and locally Lipschitz functions
Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and then f ○ g ∈ for every if and only if f is continuous on I, where stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes in Sierpiński’s classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally Lipschitz...
Simple continuity and cliquishness
Simplexe mit streng monotoner Entropiezahlenfolge
Simplicial approximation of small multifunctions
Simplicial maps which stabilize to near-homeomorphisms
Singular arc-like continua [Book]
Singularities and equicontinuity of certain families of set-valued mappings
In the present paper we establish an abstract principle of condensation of singularities for families consisting of set-valued mappings. By using it as a basic tool, the condensation of the singularities and the equicontinuity of certain families of generalized convex set-valued mappings are studied. In particular, a principle of condensation of the singularities of families of closed convex processes is derived. This principle immediately yields the uniform boundedness theorem stated in [1, Theorem...
Six theorems about injective metric spaces.
Size levels for arcs
We determine the size levels for any function on the hyperspace of an arc as follows. Assume Z is a continuum and consider the following three conditions: 1) Z is a planar AR; 2) cut points of Z have component number two; 3) any true cyclic element of Z contains at most two cut points of Z. Then any size level for an arc satisfies 1)-3) and conversely, if Z satisfies 1)-3), then Z is a diameter level for some arc.
Skeletal maps and I-favorable spaces
Skeletally Dugundji spaces
We introduce and investigate the class of skeletally Dugundji spaces as a skeletal analogue of Dugundji space. Our main result states that the following conditions are equivalent for a given space X: (i) X is skeletally Dugundji; (ii) every compactification of X is co-absolute to a Dugundji space; (iii) every C*-embedding of the absolute p(X) in another space is strongly π-regular; (iv) X has a multiplicative lattice in the sense of Shchepin [Shchepin E.V., Topology of limit spaces with uncountable...
Skorohod's spaces
Slightly -continuous functions.
Small retractions of smooth dendroids onto trees