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 ) .
A strengthening of the Čech–Pospišil theorem
Abstract We prove the following result: If in a compact space X there is a λ-branching family of closed sets then X cannot be covered by fewer than λ many discrete subspaces. (A family of sets F is λ-branching iff | F | λ but one can form λ many pairwise disjoint intersections of subfamilies of F .) The proof is based on a recent, still unpublished, lemma of G. Gruenhage. As a consequence, we obtain the following strengthening of the well-known Cech–Pospisil theorem: If X a is compact T 2 space such that all points x ∈ X have character χ ( x , X ) ⩾ κ then X cannot be covered by fewer than 2 κ many discrete subspaces.
Remotely-Piloted Helicopter Citrus Yield Map Estimation
A remotely-piloted Gas Xcell mini-helicopter with GPS, wireless Ethernet, and a machinevision system was flown over orange trees to take overhead images. The individual tree images were processed to estimate the number of orange pixels in the images. The fruit yield of individualtrees at harvest a month later had an r-squared value of 0.373 with the number of orange pixels onthe irrigated tree images. While it was demonstrated that a remotely-piloted mini-helicopter could beused to acquire useful overhead images of citrus trees, substantial further research is needed toimprove the accuracy of estimates of individual tree yields or health.