On Paracompact Spaces1
نویسندگان
چکیده
1. Stone2 has proved that a space is fully normal 7\ if and only if it is paracompact T2. If throughout his proof T\ is deleted and T2 is replaced by T4 (normality), we obtain that a space is fully normal if and only if it is paracompact TV In this note we prove that 7*2 can also be replaced by any one of the following: T2. Every two points with disjoint closures have disjoint neighborhoods. Ti. For each point x and neighborhood U of x there is a neighborhood V of x whose closure is contained in U. LTi. Every point has a neighborhood whose closure is normal. Moreover, we shall also study a natural decomposition of a space with certain properties and prove that a space is paracompact T4 if and only if it has a retract which is paracompact T2 and meets every non-null closed set. The following terminology will be used. Property *. If x, y, z are any three points such that xC\y^0 and xC\Zt£0, then yC\Zi£0. Property **. Every point has a compact closure. We remark that
منابع مشابه
Paracompactness on supra topological spaces
In this article, we present the concept of supra paracompact spaces and study its basic properties. We elucidate its relationship with supra compact spaces and prove that the property of being a supra paracompact space is weakly hereditary and topological properties. Also, we provide some examples to show some results concerning paracompactness on topology are invalid on supra topology. Finally...
متن کاملB Paracompact Does Not Imply B1 Paracompact1
As is well known the category of paracompact spaces is important in algebraic topology and the theory of fiber spaces. The following question arises naturally. If a space (Hausdorff space) B is paracompact, is the space of paths B1 (I = [0, l]), with, of course, the compact-open topology, also paracompact? The following simple example answers the question in the negative. Let X denote the set o...
متن کاملStrongly base-paracompact spaces
A space X is said to be strongly base-paracompact if there is a basis B for X with |B| = w(X) such that every open cover of X has a star-finite open refinement by members of B. Strongly paracompact spaces which are strongly base-paracompact are studied. Strongly base-paracompact spaces are shown have a family of functions F with cardinality equal to the weight such that every open cover has a l...
متن کاملMaps Having the Bula Property
Every open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed sets F0, F1 ⊂ X , with f(F0) = Y = f(F1), provided all fibers of f are infinite and C∗-embedded in X . Applications are demonstrated for the existence of “disjoint” usco multiselections of set-valued l.s.c. mappings defined on paracompact C-spaces, and for special type of factorizations of open co...
متن کامل