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Register a new userStrongly sequentially separable function spaces, via selection principles
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A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continous and Borel real-valued functions on Tychonoff spaces, with the topology of pointwise convergence. Our results solve a problem stated by Gartside, Lo, and Marsh.
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We have defined almost separable space. We show that like separability, almost separability is $c$ productive and converse also true under some restrictions. We establish a Baire Category theorem like result in Hausdorff, Pseudocompacts spaces. We investigate few relationships among separability, almost separability, sequential separability, strongly sequential separability.
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 products of general topological spaces with Mengers covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming CH{}, every productively Lindelof space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.
We prove that each metrizable space (of cardinality less or equal to continuum) has a (first countable) uniform Eberlein compactification and each scattered metrizable space has a scattered hereditarily paracompact compactification. Each compact scattered hereditarily paracompact space is uniform Eberlein and belongs to the smallest class of compact spaces, that contain the empty set, the singleton, and is closed under producing the Aleksandrov compactification of the topological sum of a family of compacta from that class.
As proved in [16], for a Tychonoff space $X$, a locally convex space $C_{p}(X)$ is distinguished if and only if $X$ is a $Delta$-space. If there exists a linear continuous surjective mapping $T:C_p(X) to C_p(Y)$ and $C_p(X)$ is distinguished, then $C_p(Y)$ also is distinguished [17]. Firstly, in this paper we explore the following question: Under which conditions the operator $T:C_p(X) to C_p(Y)$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $C_p([1,alpha])$, where $alpha$ is a countable ordinal number. A complete characterization of all $Y$ which admit a linear continuous surjective mapping $T:C_p([1,alpha]) to C_p(Y)$ is given. We also observe that for every countable ordinal $alpha$ all closed linear subspaces of $C_p([1,alpha])$ are distinguished, thereby answering an open question posed in [17]. Using some properties of $Delta$-spaces we prove that a linear continuous surjection $T:C_p(X) to C_k(X)_w$, where $C_k(X)_w$ denotes the Banach space $C(X)$ endowed with its weak topology, does not exist for every infinite metrizable compact $C$-space $X$ (in particular, for every infinite compact $X subset mathbb{R}^n$).
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