On the metrizability of certain spaces with open $(G)$
Keywords:
paracompact, open (G), developable, K-like, point-countable baseAbstract
Denote the class of all compact spaces by $\mathbb{C}$. Let $\mathbb{DC}$ be the class of all spaces for which there exists a discrete cover $\{X_i : i \in I\}$ with $\{X_i : i \in I\} \subset C$. We show that every submetacompact $\mathbb{DC}$-like Hausdorff space with open $(G)$ is developable with a point-countable base. If $X$ is a collectionwise normal $\mathbb{DC}$-like space with open $(G)$ (or a point-countable base), then $X$ is metrizable. If $X$ is a Hausdorff space with open $(G)$ (or a point-countable base) and it can be expressed as a union of countably many closed locally compact subspaces, then it is developable with a point-countable base. Every subparacompact $\mathbb{C}$-scattered Hausdorff space satisfying open $(G)$ is developable with a point-countable base. If $X$ is a paracompact $\mathbb{C}$-scattered Hausdorff space with open $(G)$ (or a point-countable base), then $X$ is metrizable. If $X$ is a monotonically normal $\mathbb{C}$-scattered space with open $(G)$ (or a point-countable base), then $X$ is metrizable. If a subspace $X$ of $\omega_1$ satisfies $(G)$, then $X$ is metrizable.
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