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Register a new userOn topological groups admitting a base at identity indexed with $omega^omega$
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A topological group $G$ is said to have a local $omega^omega$-base if the neighbourhood system at identity admits a monotone cofinal map from the directed set $omega^omega$. In particular, every metrizable group is such, but the class of groups with a local $omega^omega$-base is significantly wider. The aim of this article is to better understand the boundaries of this class, by presenting new examples and counter-examples. Ultraproducts and non-arichimedean ordered fields lead to natural families of non-metrizable groups with a local $omega^omega$-base which nevertheless are Baire topological spaces. More examples come from such constructions as the free topological group $F(X)$ and the free Abelian topological group $A(X)$ of a Tychonoff (more generally uniform) space $X$, as well as the free product of topological groups. We show that 1) the free product of countably many separable topological groups with a local $omega^omega$-base admits a local $omega^omega$-base; 2) the group $A(X)$ of a Tychonoff space $X$ admits a local $omega^omega$-base if and only if the finest uniformity of $X$ has a $omega^omega$-base; 3) the group $F(X)$ of a Tychonoff space $X$ admits a local $omega^omega$-base provided $X$ is separable and the finest uniformity of $X$ has a $omega^omega$-base.
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The concept of topological gyrogroups is a generalization of a topological group. In this work, ones prove that a topological gyrogroup G is metrizable iff G has an {omega}{omega}-base and G is Frechet-Urysohn. Moreover, in topological gyrogroups, every (countably, sequentially) compact subset being strictly (strongly) Frechet-Urysohn and having an {omega}{omega}-base are all weakly three-space properties with H a closed L-subgyrogroup
A locally convex space (lcs) $E$ is said to have an $omega^{omega}$-base if $E$ has a neighborhood base ${U_{alpha}:alphainomega^omega}$ at zero such that $U_{beta}subseteq U_{alpha}$ for all $alphaleqbeta$. The class of lcs with an $omega^{omega}$-base is large, among others contains all $(LM)$-spaces (hence $(LF)$-spaces), strong duals of distinguished Frechet lcs (hence spaces of distributions $D(Omega)$). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an $omega^{omega}$-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $omega^{omega}$-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional space $varphi$ endowed with the finest locally convex topology has an $omega^omega$-base but contains no infinite-dimensional compact subsets. It turns out that $varphi$ is a unique infinite-dimensional locally convex space which is a $k_{mathbb{R}}$-space containing no infinite-dimensional compact subsets. Applications to spaces $C_{p}(X)$ are provided.
A topological space $X$ is defined to have an $omega^omega$-base if at each point $xin X$ the space $X$ has a neighborhood base $(U_alpha[x])_{alphainomega^omega}$ such that $U_beta[x]subset U_alpha[x]$ for all $alphalebeta$ in $omega^omega$. We characterize topological and uniform spaces whose free (locally convex) topological vector spaces or free (Abelian or Boolean) topological groups have $omega^omega$-bases.
We study the relations between a generalization of pseudocompactness, named $(kappa, M)$-pseudocompactness, the countably compactness of subspaces of $beta omega$ and the pseudocompactness of their hyperspaces. We show, by assuming the existence of $mathfrak c$-many selective ultrafilters, that there exists a subspace of $beta omega$ that is $(kappa, omega^*)$-pseudocompact for all $kappa<mathfrak c$, but $text{CL}(X)$ isnt pseudocompact. We prove in ZFC that if $omegasubseteq Xsubseteq betaomega$ is such that $X$ is $(mathfrak c, omega^*)$-pseudocompact, then $text{CL}(X)$ is pseudocompact, and we further explore this relation by replacing $mathfrak c$ for some small cardinals. We provide an example of a subspace of $beta omega$ for which all powers below $mathfrak h$ are countably compact whose hyperspace is not pseudocompact, we show that if $omega subseteq X$, the pseudocompactness of $text{CL}(X)$ implies that $X$ is $(kappa, omega^*)$-pseudocompact for all $kappa<mathfrak h$, and provide an example of such an $X$ that is not $(mathfrak b, omega^*)$-pseudocompact.
Known and new results on free Boolean topological groups are collected. An account of properties which these groups share with free or free Abelian topological groups and properties specific of free Boolean groups is given. Special emphasis is placed on the application of set-theoretic methods to the study of Boolean topological groups.
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