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 by discrete and closed discrete sets
Abstract Say that a cardinal number κ is small relative to the space X if κ Δ ( X ) , where Δ ( X ) is the least cardinality of a non-empty open set in X . We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ -space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.
Monitoring and Optimization of Internal Grinding Process
Summary The grinding process is influenced by many factors such as grinding wheel characteristics, dressing conditions and, course, grinding conditions. Therefore, it is difficult to perform the grinding operation in an optimum state. In this study, in-process monitoring methods using power and acoustics emission sensors are proposed to detect malfunctions in the internal grinding process. In addition, a new internal grinding cycle featuring rapid indeed is proposed to minimize the grinding cycle time.