The main results concern commutativity of Hewitt-Nachbin realcompactification or Dieudonné completion with products of topological groups. It is shown that for every topological group $G$ that is not Dieudonné complete one can find a Dieudonné complete group $H$ such that the Dieudonné completion of $G\times H$ is not a topological group containing $G\times H$ as a subgroup. Using Korovin’s construction of $G_{\delta }$-dense orbits, we present some examples showing that some results on topological groups are not valid for semitopological...

Let $X$ be a zero-dimensional space and $C_c\left(X\right)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_c^K\left(X\right)$ (resp., $C_c^{\psi }\left(X\right))$ the set of all functions in $C_c\left(X\right)$ with compact (resp., pseudocompact) support. First, we observe that $C_c^K\left(X\right)=O_c^{{\beta }_{0}X\setminus X}$ (resp., $C_c^{\psi }\left(X\right)=M_c^{{\beta }_{0}X\setminus {\upsilon }_{0}X}$), where ${\beta }_{0}X$ is the Banaschewski compactification of $X$ and ${\upsilon }_{0}X$ is the $\mathbb{N}$-compactification of $X$. This implies that for an $\mathbb{N}$-compact space $X$, the intersection of all free maximal ideals in $C_c\left(X\right)$ is equal to $C_c^K\left(X\right)$, i.e., $M_c^{{\beta }_{0}X\setminus X}=C_c^K\left(X\right)$. By applying methods of functionally...

Two classes of spaces are studied, namely locally realcompact spaces and HN-complete spaces, where the latter class is introduced in the paper. Both of these classes are superclasses of the class of realcompact spaces. Invariance with respect to subspaces and products of these spaces are investigated. It is shown that these two classes can be characterized by demanding that certain equivalences hold between certain classes of Baire measures or by demanding that certain classes of Baire measures...

In this short article we answer the question posed in Ghadermazi M., Karamzadeh O.A.S., Namdari M., On the functionally countable subalgebra of $C\left(X\right)$, Rend. Sem. Mat. Univ. Padova 129 (2013), 47–69. It is shown that $C_c\left(X\right)$ is isomorphic to some ring of continuous functions if and only if ${\upsilon }_{0}X$ is functionally countable. For a strongly zero-dimensional space $X$, this is equivalent to say that $X$ is functionally countable. Hence for every $P$-space it is equivalent to pseudo-${\aleph }_0$-compactness.

Blum and Swaminathan [Pacific J. Math. 93 (1981), 251–260] introduced the notion of $ℬ$-fixedness for set-valued mappings, and characterized realcompactness by means of continuous selections for Tychonoff spaces of non-measurable cardinal. Using their method, we obtain another characterization of realcompactness, but without any cardinal assumption. We also characterize Dieudonné completeness and Lindelöf property in similar formulations.

A well known theorem of W.W. Comfort and K.A. Ross, stating that every pseudocompact group is $C$-embedded in its Weil completion [5] (which is a compact group), is extended to some new classes of topological groups, and the proofs are very transparent, short and elementary (the key role in the proofs belongs to Lemmas 1.1 and 4.1). In particular, we introduce a new notion of canonical uniform tightness of a topological group $G$ and prove that every $G_{\delta }$-dense subspace $Y$ of a topological group $G$, such...