Real-Analytic Non-Integrable Functions on the Plane with Equal Iterated Integrals

In this note, a vector space of real-analytic functions on the plane is explicitly constructed such that all its nonzero functions are non-integrable but yet their two iterated integrals exist as real numbers and coincide. Moreover, it is shown that this vector space is dense in the space of all real continuous functions on the plane endowed with the compact-open topology.

On first countable quasitopological homotopy groups

If q: X → Y is a quotient map, then, in general, q × q: X × X → Y × Y may fail to be a quotient map. In this paper, by reviewing the concept of homotopy groups and quotient maps, we find under which conditions the map q × q is a quotient map, where q: Ω n (X, x 0) → πn (X, x 0), is the natural quotient map from the nth loop space of (X, x 0), Ω n (X, x 0), with compact-open topology to the quasitopological nth homotopy group πn (X, x 0). Ultimately, using these results, we found some properties of first countable homotopy groups.

On the continuity of functors of the type C(X, Y)

We consider the category P, the objects of which are pairs of topological spaces (X, Y). Each such pair (X, Y) is assigned the space of continuous maps Cτ(X, Y) with some topology τ. By imposing some restrictions on objects and morphisms of category P, we define a subcategory K ⊂ P, for which the above map is a functor from K to the category Top of topological spaces and continuous maps. The following question is investigated. What are the additional conditions on K, under which the above functor is continuous? Along the way the problem of finding the limit of the inverse spectrum in the category P is solved. We show, that it reduces to finding the limits of the corresponding direct spectrum and inverse spectrum in the category Top. Point convergence topology, compact-open topology and graph topology are considered as the topology τ.

On the group of diffeomorphisms of foliated manifolds

Now the foliations theory is intensively developing branch of modern differential geometry, there are numerous researches on the foliation theory. The purpose of our paper is study the structure of the group DiffF(M) of diffeomorphisms and the group IsoF(M) of isometries of foliated manifold (M,F). It is shown the group DiffF(M) is closed subgroup of the group Diff(M) of diffeomorphisms of the manifold M in compact-open topology and also it is proven the group IsoF(M) is Lie group. It is introduced new topology on DiffF(M) which depends on foliation F and called F- compact open topology. It's proven that some subgroups of the group DiffF(M) are topological groups with F-compact open topology.

Selection principles and games in bitopological function spaces

For a Tychonoff space X, we denote by (C(X), ?k ?p) the bitopological space of all real-valued continuous functions on X, where ?k is the compact-open topology and ?p is the topology of pointwise convergence. In the papers [6, 7, 13] variations of selective separability and tightness in (C(X),?k,?p) were investigated. In this paper we continue to study the selective properties and the corresponding topological games in the space (C(X),?k,?p).

A Characterization of the Existence of a Fundamental Bounded Resolution for the Space Cc(X) in Terms of X

We characterize in terms of the topology of a Tychonoff space X the existence of a bounded resolution for CcX that swallows the bounded sets, where CcX is the space of real-valued continuous functions on X equipped with the compact-open topology.

Selection principles in function spaces with the compact-open topology

For a Tychonoff space X, we denote by Ck(X) the space of all real-valued continuous functions on X with the compact-open topology. A subset A ? X is said to be sequentially dense in X if every point of X is the limit of a convergent sequence in A. In this paper, the following properties for Ck(X) are considered. S1(S,S)=> Sfin(S,S) => Sfin(S,D) <=S1(S,D) S1(D,S) => Sfin(D,S) => Sfin(D,D) <= S1(D,D) For example, a space Ck(X) satisfies S1(S,D) (resp., Sfin(S,D)) if whenever (Sn : n ? N) is a sequence of sequentially dense subsets of Ck(X), one can take points fn ? Sn (resp., finite Fn ? Sn) such that {fn : n ? N} (resp.,U {Fn : n ? Ng) is dense in Ck(X). Other properties are defined similarly. In [22], we obtained characterizations these selection properties for Cp(X). In this paper, we give characterizations for Ck(X).