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Register a new userSome observations on the Baireness of C_k(X) for a locally compact space X
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Franklin D. Tall
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We prove some consistency results concerning the Moving Off Property for locally compact spaces and thus the question of whether their function spaces are Baire.
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We prove that for a stratifiable scattered space $X$ of finite scattered height, the function space $C_k(X)$ endowed with the compact-open topology is Baire if and only if $X$ has the Moving Off Property of Gruenhage and Ma. As a byproduct of the proof we establish many interesting Baire category properties of the function spaces $C_k(X,Y)={fin C_k(X,Y):f(X)subset{*_Y}}$, where $X$ is a topological space, $X$ is the set of non-isolated points of $X$, and $Y$ is a topological space with a distinguished point $*_Y$.
Given a class $mathcal P$ of Banach spaces, a locally convex space (LCS) $E$ is called {em multi-$mathcal P$} if $E$ can be isomorphically embedded into a product of spaces that belong to $mathcal P$. We investigate the question whether the free locally convex space $L(X)$ is strongly nuclear, nuclear, Schwartz, multi-Hilbert or multi-reflexive. If $X$ is a Tychonoff space containing an infinite compact subset then, as it follows from the results of cite{Aus}, $L(X)$ is not nuclear. We prove that for such $X$ the free LCS $L(X)$ has the stronger property of not being multi-Hilbert. We deduce that if $X$ is a $k$-space, then the following properties are equivalent: (1) $L(X)$ is strongly nuclear; (2) $L(X)$ is nuclear; (3) $L(X)$ is multi-Hilbert; (4) $X$ is countable and discrete. On the other hand, we show that $L(X)$ is strongly nuclear for every projectively countable $P$-space (in particular, for every Lindelof $P$-space) $X$. We observe that every Schwartz LCS is multi-reflexive. It is known that if $X$ is a $k_omega$-space, then $L(X)$ is a Schwartz LCS cite{Chasco}, hence $L(X)$ is multi-reflexive. We show that for any first-countable paracompact (in particular, metrizable) space $X$ the converse is true, so $L(X)$ is multi-reflexive if and only if $X$ is a $k_omega$-space, equivalently, if $X$ is a locally compact and $sigma$-compact space. Similarly, we show that for any first-countable paracompact space $X$ the free abelian topological group $A(X)$ is a Schwartz group if and only if $X$ is a locally compact space such that the set $X^{(1)}$ of all non-isolated points of $X$ is $sigma$-compact.
We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of omega_1 is hereditarily paracompact.
In his seminal work cite{pal:61}, R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results well known for compact group actions. In particular, we prove that if $H$ is a compact subgroup of a locally compact group $G$ and $S$ is a small (in the sense of Palais) $H$-slice in a proper $G$-space, then the action map $Gtimes Sto G(S)$ is open. This is applied to prove that the slicing map $f_S:G(S)to G/H$ is continuos and open, which provides an external characterization of a slice. Also an equivariant extension theorem is proved for proper actions. As an application, we give a short proof of the compactness of the Banach-Mazur compacta.
Let $X$ and $Y$ be topological spaces. Let $C$ be a path-connected closed set of $Xtimes Y$. Suppose that $C$ is locally direct product, that is, for any $(a,b)in Xtimes Y$, there exist an open set $U$ of $X$, an open set $V$ of $Y$, a subset $I$ of $U$ and a subset $J$ of $V$ such that $(a,b) in Utimes V$ and $$Ccap (Utimes V)=Itimes J$$ hold. Then, in this memo, we show that $C$ is globally so, that is, there exist a subset $A$ of $X$ and a subset $B$ of $Y$ such that $$C=Atimes B$$ holds. The proof is elementary. Here, we note that one might be able to think of a (perhaps, open) similar problem for a fiber product of locally trivial fiber spaces, not just for a direct product of topological spaces. In Appendix, we mentioned a simple example of a $C([0,1];mathbb R)$-manifold that cannot be embedded in the direct product $(C([0,1];mathbb R))^n$ as a $C([0,1];mathbb R)$-submanifold. In addition, we introduce the concept of topological 2-space, which is locally the direct product of topological spaces and an analog of homotopy category for topological 2-space. Finally, we raise a question on the existence of an $mathbb R^n$-Morse function and the existence of an $mathbb R^n$-immersion in a finite-dimensional $mathbb R^n$-Euclidean space. Here, we note that the problem of defining the concept of an $mathbb R^n$-handle body may also be considered.
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