| The main purpose of this paper is to prove the following two theorems, an order hereditary closure preserving sum theorem and an hereditary theorem: (1) If a topological property $mathcal{P}$ satisfies $(sum')$ and is closed hereditary, and if $mathcal{V}$ is an order hereditary closure preserving open cover of $X$ and each $V inmathcal{V}$ is elementary and possesses $mathcal{P}$, then $X$ possesses $mathcal{P}$. (2) Let a topological property $mathcal{P}$ satisfy $(sum')$ and $(beta),$ and be closed hereditary. Let $X$ be a topological space which possesses $mathcal{P}$. If every open subset $G$ of $X$ can be written as an order hereditary closure preserving (in $G$) collection of elementary sets, then every subset of $X$ possesses $mathcal{P}$. |
Keywords
elementary, order hereditary closure preserving, sum theorem
Math Review Classification
Primary 54D20
Last Updated
31/03/2006
Length
10 pages
Availability
This article is available in: