Notes

Description: Definitions from Manka [1988a] and Manka [1988b]

Content:

Definitions from Manka [1988a] and Manka [1988b]

Definition:  Suppose that \((X,\le )\) is a partial ordering

  1. A set \({\cal A} \subseteq  {\cal P}(X)\) is supremal in \((X,\le )\) if
    1. \(\forall S\in{\cal A}\) and for every upper bound \(y\) of \(S\), \(S\cup\{y\}\in{\cal A}\) and
    2. \(\forall  S \in  {\cal A}\) every initial segment  of  \(S\) is  in \({\cal A}\).
  2. If \({\cal A}\) is supremal in \((X,\le)\) and \(\sigma :{\cal A} \rightarrow X\) then \(\sigma\) is a sup function iff
    1. \(\forall S\in{\cal A}\), \(\sigma(S)\) is an upper bound for \(S\)  and
    2. \(\forall S\in{\cal A}\), \(\sigma (S)\) is the greatest element of \(S\) if there is one.
  3. \((X,\le )\) is directed if each two element subset of \(X\) has an upper bound.

Definition:If \((X,\rho )\) is a metric space and \(\phi: X \rightarrow{\Bbb R}\), then \(\phi\) is upper semi-continuous iff for every convergent  sequence \(\{x_{n}\}\) in \((X,\rho )\) and for all \(r\in{\Bbb R}\), the inequality \(\phi(x_{n})\ge r\) for all \(n\in\omega\) implies \(\phi (\lim_{} x_{n})\ge r\).

Definition:If \((X,\le)\) is a directed, partially ordered set and \(\tau\) is a topology on \(X\), then \((X,\le)\) is upper-semi continuous with respect to \(\tau\) if for every \(x_0\) in \(X\), \(\{y\in X: x_0\le y\}\) is closed.

Howard-Rubin number: 38

Type: Summary of definitions

Back