Mal'tsev and retral spaces

A space X is Mal'tsev if there exists a continuous map M: X3 → X such that M(x, y, y) = x = M(y, y, x). A space X is retral if it is a retract of a topological group. Every retral space is Mal'tsev. General methods for constructing Mal'tsev and retral spaces are given. An example of a Mal'tsev space which is not retral is presented. An example of a Lindelöf topological group with cellularity the continuum is presented. Constraints on the examples are examined

Mal'tsev and retral spaces

A space X is Mal'tsev if there exists a continuous map M: X3 → X such that M(x, y, y) = x = M(y, y, x). A space X is retral if it is a retract of a topological group. Every retral space is Mal'tsev. General methods for constructing Mal'tsev and retral spaces are given. An example of a Mal'tsev space which is not retral is presented. An example of a Lindelöf topological group with cellularity the continuum is presented. Constraints on the examples are examined.Copyright 1997 Elsevier B.V. All rights reserved. Re-use of this article is permitted in accordance with the Terms and Conditions set out at http://www.elsevier.com/open-access/userlicense/1.0

Cellularity and the index of narrowness in topological groups

summary:We study relations between the cellularity and index of narrowness in topological groups and their $G_\delta$-modifications. We show, in particular, that the inequalities $\operatorname{in} ((H)_\tau)\le 2^{\tau\cdot \operatorname{in} (H)}$ and $c((H)_\tau)\leq 2^{2^{\tau\cdot \operatorname{in} (H)}}$ hold for every topological group $H$ and every cardinal $\tau\geq \omega$, where $(H)_\tau$ denotes the underlying group $H$ endowed with the $G_\tau$-modification of the original topology of $H$ and $\operatorname{in} (H)$ is the index of narrowness of the group $H$. Also, we find some bounds for the complexity of continuous real-valued functions $f$ on an arbitrary $\omega$-narrow group $G$ understood as the minimum cardinal $\tau\geq \omega$ such that there exists a continuous homomorphism $\pi\colon G\to H$ onto a topological group $H$ with $w(H)\leq \tau$ such that $\pi\prec f$. It is shown that this complexity is not greater than $2^{2^\omega }$ and, if $G$ is weakly Lindelöf (or $2^\omega$-steady), then it does not exceed $2^\omega$

Conway’s Question: The Chase for Completeness

We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone–Čech compactification β X (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory Conw of the category Tych of Tychonoff spaces. Replacing sequential closure by the general notion of a closure operator C, we introduce analogously the subcategory Conw C of C-Conway spaces, that turns out to be again a bireflective subcategory of Tych. We show that every bireflective subcategory of Tych can be presented in this way by building a Galois connection between bireflective subcategories of Tych and closure operators of Top finer than the Kuratowski closure. Other levels of completeness are considered for the (underlying topological spaces of) topological groups. A topological group G is sequentially complete if it is sequentially closed in its Raĭkov completion . The sequential completeness for topological groups is stronger than Conway’s property, although they coincide in some classes of topological groups, for example: free (Abelian) topological groups, pseudocompact groups, etc