In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fréchet-Urysohn fan Sω in a topological group G admitting a functorial embedding [0, 1] ⊂ G. The latter means that each autohomeomorphism of [0, 1] extends to a continuous homomorphism of G. This implies that many natural free topological group constructions (e.g. the constructions of the Markov free t...

We prove that, for each countable ordinal ξ ≥ 1, there exist some Σ0ξ-complete ω-powers, and some Π0ξ-complete ω-powers, extending previous works on the topological complexity of ω-powers [Fin01, Fin03, Fin04, Lec01, Lec05, DF06]. We prove effective versions of these results; in particular, for each recursive ordinal ξ < ω 1 there exist some recursive sets A ⊆ 2 such that A∞ ∈ Π 0ξ \Σ 0 ξ (resp...

We prove that, for each countable ordinal ξ ≥ 1, there exist some Σ0ξ-complete ω-powers, and some Π0ξ-complete ω-powers, extending previous works on the topological complexity of ω-powers [Fin01, Fin03, Fin04, Lec01, Lec05, DF06]. We prove effective versions of these results; in particular, for each recursive ordinal ξ < ω 1 there exist some recursive sets A ⊆ 2 such that A∞ ∈ Π 0ξ \Σ 0 ξ (resp...

In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of [vdD1, 2.11], and the lifting principle we develop is a generalization of the geometric axiomatization of ...

In this article, we will be interested in the extent to which the assumption of first countability in this theorem can be weakened. Recall that a Hausdorff topological space X is Fréchet if whenever x is a limit point of A ⊆ X, there is a sequence an (n < ω) of elements of A which converges to x. This is a natural weakening of first countability which has been extensively studied in the literat...

We prove that, for each countable ordinal ξ ≥ 1, there exist some Σ0ξ-complete ω-powers, and some Π0ξ-complete ω-powers, extending previous works on the topological complexity of ω-powers [Fin01, Fin03, Fin04, Lec01, Lec05, DF06]. We prove effective versions of these results; in particular, for each recursive ordinal ξ < ω 1 there exist some recursive sets A ⊆ 2 such that A∞ ∈ Π 0ξ \Σ 0 ξ (resp...

Let φn be an arbitrary compact topological manifold. We shall define a new topological object, Ω(t)φn , known as a lure, that alters the Minkowski Content of φn by acting on the boundary of φn, ∂φn by stretching or shrinking the boundary. After the notion of a lure has been established for compact manifolds, we shall relax the requirement that the manifold be locally Euclidean, by defining Λn t...

A simple topological graph is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once. Using a new tool developed in [12] we show that every simple topological graph on n vertices contains Ω(n 1 2 / √ log n) pairwise disjoint edges. This improves the previous lower bound of Ω(n 1 3 ) by Suk [17] and by Fulek and Ru...

In this paper, we introduce and characterize the concept of fuzzy almost generalized $e$-continuous mappings. Several interesting properties of these mappings are also given. Examples and counter examples are also given to illustrate the concepts introduced in the paper. We also introduce the concept of fuzzy $f T_{frac{1}{2}}e$-space, fuzzy $ge$-space, fuzzy regular $ge$-space and  fuzzy gener...

The complementation problem for nondeterministic automata on infinite words has numerous applications in formal verification. In particular, the language-containment problem, to which many verification problems are reduced, involves complementation. Traditional optimal complementation constructions are quite complicated and have not been implemented. Recently, we have developed an analysis tech...