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Let $p:X \to Y$ be a perfect map. Then: $(4)$ If $X$ is $2^\circ$ countable, so also is $Y$. Dugundji’s proof: Let $\{U_i\}$ be a countable basis for $X$, and let $\{V_i\}$ be the family of all finite unions of $U_i$; by chapter 2, 8.8, the family $\{V_i\}$ is countable, and we show that the o...