In 1960, Arhangel'skii gave a metrization theorem, showing that a space is metrizable if and only if it has a regular base. In the present paper, we prove two metrization theorems analogous tu Arhangel'skii's. For these two, we make use of regular k-networks and generalized regular bases, called HCP-regular bases, respectively. Next, we give a characterization of Lasnev spaces, using HCP-regular networks. Moreover, we give a partial answer to a problem concerning Lasnev spaces and regular networks, raised by Junnila and Yajima.
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In this paper it is showed that a regular and k-space with a regular k-network is metrizable, which generalized related results of A. Archangielskiĭ, H. W. Martin, M. Sakai, K. Tamano and Y. Yajima.
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