# Questions tagged [perfect-map]

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### Proof verification of exercise 7(a), section 31 of Munkres’ topology

Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}\big(\{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (a) Show that if $X$ is ...
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### Exercise 7(c), Section 31 of Munkres’ Topology

Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1} \big( \{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (c) Show that if $X$ is ...
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### Exercise 7(b), Section 31 of Munkres’ Topology

Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1} \big( \{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (b) Show that if $X$ is ...
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### Exercise 7(a), Section 31 of Munkres’ Topology

Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}\big(\{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (a) Show that if $X$ is ...
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 ...
Let $p:X\to Y$ be a closed continuous surjective map such that $p^{-1}(\{y\})$ is compact, for each $y\in Y$. (Such a map is called a perfect map) Show if $Y$ is compact, then $X$ is compact. [Hint: ...