We characterize a class of topological Ramsey spaces such that each element R of the class induces a collection {Rk}k<ω of projected spaces which have the property that every Baire set is Ramsey. Every projected space Rk is a subspace of the corresponding space of length-k approximation sequences with the Tychonoff, equivalently metric, topology. This answers a question of S. Todorcevic and gen...

In the first part of this note an elementary proof is given of the fact that algebraic functors, that is, functors induced by morphisms of Lawvere theories, have left adjoints provided that the category K in which the models of these theories take their values is locally presentable. The main focus however lies on the special cases of the underlying functor of the category Grp(K) of internal gr...

We continue from “part I” our address of the following situation. For a Tychonoff space Y, the “second epi-topology” σ is a certain topology on C(Y), which has arisen from the theory of categorical epimorphisms in a category of lattice-ordered groups. The topology σ is always Hausdorff, and σ interacts with the point-wise addition + on C(Y) as: inversion is a homeomorphism and + is separately c...

All spaces here are Tychonoff spaces. The class AE(0) consists of those spaces which are absolute extensors for compact zero-dimensional spaces. We define and study here the subclass AE(0), consisting of those spaces for which extensions of continuous functions can be chosen to have the same range. We prove these results. If each point of T ∈ AE(0) is a Gδ-point of T , then T ∈ AE(0) . These ar...

Given a topological space X, a subspace Y of X is said to be Cembedded in X if every continuous real valued function on Y has a continuous extension to all of X. The notion of C-embedding is well understood and documented, as well as that of the more general notions of C∗-embedding and z-embedding. In this article we define a generalization of C-embedding and investigate some of its interesting...

All spaces in this paper are Tychonoff. A Wallman base on a space X is a normal separating ring of closed subsets of X (see Steiner, Duke Math. J. 35 (1968), 269-276). Let T be a compact space and £ a Wallman base on T. For XCZT, define £x = {Ar)X\AE£}. Theorem 1. If X is a dense subspace of T, then T = w£x iff cItAHclrB = 0 whenever A, S£& and AC\B = 0. Theorem 2. T = w£xfor each dense XCZT if...

For the study of some typical problems in finance and economics, Žitković introduced convex compactness gave many remarkable applications. Recently, motivated by random optimization variational inequalities, Guo, et al. L0-convex compactness, developed related theory normed modules further applied it to backward stochastic equations. In this paper, we extensively L0-convexly compact sets locall...

Given a topological group G that can be embedded as subgroup into some vector space (over the field of reals) we say has invariant linear span if all spans under arbitrary embeddings spaces are isomorphic spaces. For an set A let $${{\mathbb {Z}}}^{(A)}$$ direct sum |A|-many copies discrete integers endowed with Tychonoff product topology. We show span. This answers question from paper Dikranja...

Abstract An internal characterization of the Arkhangel’skiĭ-Calbrix main theorem from [4] is obtained by showing that space $$C_{p}(X)$$ C p ( X ) continuous real-valued functions on a Tychonoff X...

The weak topology of a locally convex space (lcs) E is denoted by w. In this paper we undertake systematic study those lcs such that (E, w) (linearly) Eberlein–Grothendieck (see Definitions 1.2 and 3.1). following results obtained in our play key role: for every barrelled E, the (linearly Eberlein–Grothendieck) if only metrizable (E normable, respectively). main applications concern to continuo...