On the Order Hereditary Closure Preserving Sum Theorem

Jianhua Gong and Ivan L. Reilly


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 $mathcal{P}$.

elementary, order hereditary closure preserving, sum theorem

Math Review Classification
Primary 54D20

Last Updated

10 pages

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