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Register a new userTopological gyrogroups with Frechet-Urysohn property and 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
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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.
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.
A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. In this paper, it is proved that if $G$ is a sequential topological gyrogroup with an $omega^{omega}$-base, then $G$ has the strong Pytkeev property. Moreover, some equivalent conditions about $omega^{omega}$-base and strong Pytkeev property are given in Baire topological gyrogroups. Finally, it is shown that if $G$ is a strongly countably complete strongly topological gyrogroup, then $G$ contains a closed, countably compact, admissible subgyrogroup $P$ such that the quotient space $G/P$ is metrizable and the canonical homomorphism $pi :Grightarrow G/P$ is closed.
Topological gyrogroups, with a weaker algebraic structure without associative law, have been investigated recently. We prove that each $T_{0}$-strongly topological gyrogroup is completely regular. We also prove that every $T_{0}$-strongly topological gyrogroup with a countable pseudocharacter is submetrizable. Finally, we prove that the left coset space $G/H$ is submetrizable if $H$ is an admissible $L$-subgyrogroup of a $T_{0}$-strongly topological gyrogroup $G$.
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