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$\pi$-metrizable; weakly $\pi$-metrizable; $\pi$-base; $\sigma$-discrete $\pi$-base; $\sigma$-disjoint $\pi$-base; $d$-separable
A space $X$ is said to be $\pi$-metrizable if it has a $\sigma$-discrete $\pi$-base. The behavior of $\pi$-metrizable spaces under certain types of mappings is studied. In particular we characterize strongly $d$-separable spaces as those which are the image of a $\pi$-metrizable space under a perfect mapping. Each Tychonoff space can be represented as the image of a $\pi$-metrizable space under an open continuous mapping. A question posed by Arhangel'skii regarding if a $\pi$-metrizable topological group must be metrizable receives a negative answer.
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