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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 $ℝ$, 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...