How to construct point finite covering in collectionwise normal spaces?

How to construct point finite covering in collectionwise normal spaces

• I am actually looking for a related reference (and ideally if anyone knows the answer) on the following construction problem: Let X= $\prod_{i=1,..,n} X_{i}$ be a collectionwise normal and Hausdorff locally convex topological vector space and for every $i \in$ {1,...,n} let $X_{i}$ be just a Hausdorff locally convex topological vector space. In addition let's assume for every $i \in$ {1,...,n} an upper semicontinuous multivalued map $\phi_{i}$:X-> $P(X_{i})$, with $\phi_{i}$(x) non-empty, convex and compact subset of $X_{i}$, for every x$\in$X. Given the open covering ($V_{x})_{x\in X}$ of X (where each $V_{x}$ in an open neighborhood of x) and the assumptions above, can a point finite covering of X that is subordinated to ($V_{x})_{x\in X}$ be constructed ?

• Answer:

  It seems the following. The families $\phi_i$ are not used in the question and the condition “$X_i$ be just a Hausdorff locally convex topological vector space” is abundant. Moreover, I shall consider an open refinement (that is, a cover not indexed by elements of $X$) of the open cover $(V_x)_{x\in X}$, because each subordinated open cover $(U_x)_{x\in X}$ is not point finite even for $X=\mathbb R$, because each point finite open cover of a separable space is countable. So I reformulate the question as “Is a collectionwise normal locally convex topological vector space $X$ a weakly paracompact space?” I remark that, by Michael-Nagami Theorem [Eng, 5.3.3], each collectionwise normal weakly paracompact space is paracompact. A regular space $X$ is submetacompact (or $\theta$-refinable) if for each open cover $\mathcal U$ of $X$ there exists a sequence $\{\mathcal V_n\}$ of open reinements of $\mathcal U$ such that for each $x\in X$, there exists $n\in\omega$ such that $x$ is in only finitely many elements of $\{\mathcal V_n\}$. [Gru, 1.8]. Each regular paracompact space is submetacompact, and a collectionwise normal space $X$ is paracompact iff $X$ is submetacompact. Greg Kuperberg http://mathoverflow.net/a/8660 “I did find one general criterion that implies that a locally convex space is paracompact. According to the Encyclopedia of Mathematics, if it is Montel (which means that it is barrelled and the Heine-Borel theorem holds true for it), then it is paracompact”. Henno Brandsma http://mathoverflow.net/questions/44907/an-example-of-a-non-paracompact-tvs-over-the-reals-say#comment108716_44929 that a space $C p (X)$ of continuous functions on a Tychonov space $X$ in the pointwise topology is paracompact iff it is normal. References [Eng] Ryszard Engelking. http://webfile.ru/efb452cf7fb36adbe54307b44eb8657f. Berlin, Heldermann, 1989. [Gru] Gary Gruenhage, Generalized Metric Spaces, in K. Kunen and J. E. Vaughan, http://webfile.ru/6475714, Elsevier, 1984.