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almost compact map; pseudo-open map; almost injective map; discrete family; scattered
We show that all continuous maps of a space $X$ onto second countable spaces are pseudo-open if and only if every discrete family of nonempty $G_\delta $-subsets of $X$ is finite. We also prove under CH that there exists a dense subspace $X$ of the real line $\Bbb R$, such that every continuous map of $X$ is almost injective and $X$ cannot be represented as $K\cup Y$, where $K$ is compact and $Y$ is countable. This partially answers a question of V.V. Tkachuk in [Tk]. We show that for a compact $X$, all continuous maps of $X$ onto second countable spaces are almost injective if and only if it is scattered. We give an example of a non-compact space $Z$ such that every continuous map of $Z$ onto a second countable space is almost injective but $Z$ is not scattered.
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