Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [perfect-map]

Apt for questions related to perfect map which is a particular kind of continuous function between topological spaces that are weaker than homeomorphisms, but strong enough to preserve some topological properties such as local compactness that are not always preserved by continuous maps.

Let $X$ and $Y$ be topological spaces and let $p$ be a map from $X$ to $Y$ that is continuous, closed, surjective and such that each fiber $p^{-1}(y)$ is compact relative to $X$ for each $y$ in $Y$. Then $p$ is known as a perfect map.

To know more on this check this link.

6 questions
9
votes
1 answer

Mysior plane is not realcompact

Let $X = \mathbb{R}^2$ with $(x, y)\in X$ for $y\neq 0$ isolated and $(x, 0)$ having neighbourhood basis of the form $$U_n(x) = \{(x, y) : y\in (-1/n, 1/n)\}\cup \{(x+y+1, y) : 0 < y < 1/n\}\cup \{(x+\sqrt{2}+y, -y) : 0 < y < 1/n\}.$$ The space $X$…
Jakobian
  • 15,280
2
votes
1 answer

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 Hausdorff, then so is $Y$. My…
user264745
  • 4,595
2
votes
1 answer

Exercise 12, Section 26 of Munkres’ Topology

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: If $U$ is open set containing $p^{-1}(\{y\})$,…
user264745
  • 4,595
1
vote
0 answers

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 locally compact, then so is $Y$. My attempt:…
user264745
  • 4,595
0
votes
0 answers

Inverse image of closure under perfect open mapping.

Let $f$ be an open and perfect mapping from $X$ to $Y$. Let $A \subset Y$ and let $y \in \overline{A}$, then whether it is true that $f^{-1}(y) \subset \overline{f^{-1}(A)}$. I have assumed that $y \in \overline{A} \setminus A$. Let $x = f^{-1}(y)$…
Sumit Mittal
  • 481
0
votes
1 answer

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 Hausdorff, then so is $Y$. My attempt: It’s easy…
user264745
  • 4,595