COMPACTIFICATIONS OF A FIXED SET

Totally Lindelöf and totally ω-narrow paratopological groups

On the local connectedness of frames

The path component space of a topological space $X$ is the quotient space $\pi_0(X)$ whose points are the path components of $X$. We show that every Tychonoff space $X$ is the path-component space of a Tychonoff space $Y$ of weight $w(Y)=w(X)$ such that the natural quotient map $Y\to \pi_0(Y)=X$ is a perfect map. Hence, many topological properties of $X$ transfer to $Y$. We apply this result to construct a compact space $X\subset \mathbb{R}^3$ for which the fundamental group $\pi_1(X,x_0)$ is an uncountable, cosmic, $k_{\omega}$-topological group but for which the canonical homomorphism $\psi:\pi_1(X,x_0)\to \check{\pi}_1(X,x_0)$ to the first shape homotopy group is trivial.

Notes on countable tightness of the subspaces of free (Abelian) topological groups

The general concept of cleavability of a topological space

Some aspects of dimension theory for topological groups

A subset Y of the dual closed unit ball of a Banach space E is called a Rainwater set for E if every bounded sequence of E that converges pointwise on Y converges weakly in E. In this paper, topological properties of Rainwater sets for the Banach space of the real‐valued continuous and bounded functions defined on a completely regular space X equipped with the supremum‐norm are studied. This applies to characterize the weak K‐analyticity of in terms of certain Rainwater sets for . Particularly, we show that is weakly K‐analytic if and only if there exists a Rainwater set Y for such that is both K‐analytic and angelic, where denotes the topology on of the pointwise convergence on Y. For the case when X is compact, one gets classic Talagrand's theorem. As an application we show that if X is a compact space and Y is a ‐dense subspace, then X is Talagrand compact, i.e., is K‐analytic, if and only if the space is K‐analytic.

The topological properties of q-spaces in free topological groups

Let B be a base for a nowhere locally compact Tychonoff space X and let bX be a compactification of X. Then the following two statements hold: (1) The remainder bX \ X of X is pseudocompact if and only if for any countable infinite subfamily V of B there exists an accumulation point of the family V in bX \ X. (2) If for any countable infinite subfamily V of B the set of all accumulation points of the family V in X is not a nonempty compact set of X, then bX \ X is pseudocompact. Let X = Q i?I Xi be a product space and S be a subset of X satisfying the following condition: (*) For each nonempty countable set J ? I, the projection pJ : X ? Q Q i?J Xi satisfies that pJ(S) = XJ := i?J Xi. If B is the canonical base for X and VS = {Bi ? S : i ? ?} is a countable infinite subfamily of BS = {B ? S : B ? B} such that the set F of all accumulation points of the family VS in S is nonempty, then for any a ? F there exists a countable subset J of I such that p?1 J (pJ(a)) ? S = p?1 J (pJ(a)) ? F and for any ? ? I \ J, p?(F) = X?. By the above conclusions, we can get two known results in [8]. We finally show that if X = Q i?I Xi is a product of a family {Xi : i ? I} of Tychonoff spaces such that uncountably many of them are non-compact and Y is a dense subspace of X, then for every compactification bY of Y the remainder bY\Y is pseudocompact.

S2 and the Fréchet property of free topological groups

Fréchet–Urysohn subspaces of free topological groups III

A Tychonoff space X is called (sequentially) Ascoli if every compact subset (resp. convergent sequence) of $$C_k(X)$$ is equicontinuous, where $$C_k(X)$$ denotes the space of all real-valued continuous functions on X endowed with the compact-open topology. The classical Ascoli theorem states that each compact space is Ascoli. We show that a pseudocompact space X is Asoli iff it is sequentially Ascoli iff it is selectively $$\omega $$ -bounded. The class of selectively $$\omega $$ -bounded spaces is studied.

1. In view of the recent work on topological groups it is natural to consider the situation which arises when such groups act as transformation groups on various types of spaces. Such a study is begun here from the point of view of almost periodic transformation groups, the definition of which is suggested by von Neumann's paper on almost periodic functions in a group.t Compact topological transformation groups are a special case of almost periodic transformation groups, at least for a rather wide class of spaces. The paper concerns itself chiefly with the nature of the minimal closed invariant sets of such groups. There are some results for general spaces but the main results are for Euclidean spaces and more particularly for threedimensional Euclidean spaces. One of the most interesting theorems states that if a compact one-dimensional group acts on three-space in such a way that its orbits are uniformly bounded in diameter, then every point of the space is fixed under the group, so that if such a group is to act in a nontrivial manner the diameters of its orbits must be unbounded. Under some restrictions a similar theorem is proved for one-parameter almost periodic groups. Furthermore it is shown that for this latter class of groups, many of the orbits must actually be simple closed curves if they have one-dimensional closures. 2. The group considered here will be denoted by G. It will be subjected to various conditions as the occasion demands but it will always be Abelian. In case it is the group of real numbers, it will be spoken of as a one-parameter group; in case it is the real numbers reduced modulo one, it will be spoken of as the circle group. The space on which the group acts will be denoted by R. It will be specialized in various ways, but in any case it will always be a locally compact metric space. If x and y are two points of R, the distance between them will be denoted by d(x, y).

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

In (6) we studied a topology T on the product set X × Y of two topological spaces X and Y which was defined by the requirement that each mapping from X × Y which was continuous in each variable separately was also continuous in T; we called (X × Y, T) the tensor product of X and Y, and denoted it by X ⊗ Y. Theorem (3·2) of (6) indicated that X ⊗ Y was rarely completely regular; as complete regularity is of importance in analytic problems, we consider here a ‘completely regular tensor product’ . Roughly speaking, gives a tensor product in the category of completely regular topological spaces. The categorical properties of are discussed in section 5.