Covering compacta by discrete subspaces
Abstract For any space X, denote by dis ( X ) the smallest (infinite) cardinal κ such that κ many discrete subspaces are needed to cover X. It is easy to see that if X is any crowded (i.e. dense-in-itself) compactum then dis ( X ) ⩾ m , where m denotes the additivity of the meager ideal on the reals. It is a natural, and apparently quite difficult, question whether in this inequality m could be replaced by c . Here we show that this can be done if X is also hereditarily normal. Moreover, we prove the following mapping theorem that involves the cardinal function dis ( X ) . If f : X → Y is a continuous surjection of a countably compact T 2 space X onto a perfect T 3 space Y then | { y ∈ Y : f −1 y  is countable } | ⩽ dis ( X ) .
Covering compacta by discrete and other separated sets
Article history: We show that if a space X is the union of not more than κ-many discrete subspaces, where κ is an infinite cardinal, then the same holds for any perfect image of X. It follows that a compact Hausdorff space with no isolated points can never be covered by fewer than continuum many discrete subspaces; this answers a question of I. Juhasz and J. van Mill. We also consider coverings by right-separated and left-separated subspaces. © 2008 Elsevier B.V. All rights reserved.
Metallurgical Process Engineering
The "material of choice" - steel.- Uprising of China's steel industry.- Process manufacturing industry and process engineering.- Engineering science in steel manufacturing process.- Analysis and integration of metallurgical manufacturing process.- Multi-dimensional mass flow control for metallurgical manufacturing process.- Time factors in steel manufacturing process.-Operating dynamics in the production process of steel plant.- Structure and mode of steel plant.- Steel plant and environment.