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Questions tagged [proper-maps]

For questions related to proper maps. A function between topological spaces is called proper if inverse images of compact subsets are compact.

There are several competing definitions of a "proper function". Some authors call a function $f:X\to Y$ between two topological spaces proper if the preimage of every compact set in $Y$ is compact in $X$. Other authors call a map $f$ proper if it is continuous and closed with compact fibers; that is if it is a continuous closed map and the preimage of every point in $Y$ is compact. The two definitions are equivalent if $Y$ is locally compact and Hausdorff.

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23 questions
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Factorization of maps between locally compact Hausdorff space

Let me consider a space to be locally compact, if every point has a neighborhood whose closure is compact. Consider a continuous map $f:X\to Y$ between locally compact Hausdorff spaces. Is it true that $f$ can be factored as…
gurgur dilon
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A topological Ehresmann's theorem

A proper local homeomorphism is a covering map (assuming some mild conditions on the involved spaces). I want to know about the following generalization, which I believe is false but cannot come up with a counterexample to. Suppose $f : E \to B$ is…
ronno
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Showing that $\mathbb A^n_\mathbb C\rightarrow \operatorname{Spec}\mathbb C$ is not proper.

$\newcommand{\Spec}{\operatorname{Spec}}\newcommand{\P}{\mathbb P}\newcommand{\C}{\mathbb C}\newcommand{\A}{\mathbb A}$ Let $\A^n_\C=\Spec \C[x_1,\dots,x_n]$, and let $f:\A^n_\C\rightarrow \Spec \C$ be the obvious morphism induced by the inclusion…
Chris
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Characterisation of a proper map

Let $X$ and $Y$ be topological spaces. A continuous map $F:X \rightarrow Y$ is called proper if the preimage of any compact subset in $Y$ is a compact subset of $X$. I wish to understand the definition of a proper map because it is a key ingredient…
Joseph Kwong
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Base change of proper map is proper

Let $X,B,B'$ be topological spaces. Let $\pi\colon X\to B$ be a continuous, proper map (proper in the sense of Bourbaki-proper: closed and with quasi-compact fibres). Let $f\colon B'\to B$ be continuous. Form the fibre product $X\times_{B} B'$. I…
rae306
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A necessary condition for a proper map

In Steinmetz's Rational Iteration, a holomorphic map $f$ from domain $D$ into some domain $G$ is said to be proper if there is a $k \in \mathbb{Z}^+$ such that $f:D \overset{k:1}{\longrightarrow} G$ ie. every point in $G$ has exactly $k$ preimages…
Ajin Shaji Jose
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Is every regular action proper?

Let $A:\,G\times X\to X$ be a continuos action of a topological group $G$ on a topological space $X$. Question 1: Assume that the action is regular (ie free and transitive). Does that imply that the action is proper? If not: what more do we need for…
Alex
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If a submanifold is homeomorphic to the ambient manifold but is a proper subset, then the inclusion map can't be proper

I am solving the following problem (the motivation is that the inclusion of the open unit disk in the plane is not proper): Problem: Let $M$ be a connected, non-compact topological manifold without a boundary, and $N$ be a submanifold of $M$ such…
Random
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If two proper maps are homotopic, then they are properly homotopic

Let $M$ and $N$ be smooth manifolds, and let $f$ and $g$ be proper smooth maps $M\to N$ such that there is a smooth map $F:\mathbb{R}\times M \to N$ with $F_0=f$ and $F_1=g$. Is there a proper smooth map $G:\mathbb{R}\times M \to N$ with $G_0=f$ and…
marino
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Proper base change in etale cohomology and surjectivity on Picard groups

I have the question concerning one part of the proof of proper base change in etale cohomology. At one point during the proof we have the following setup and the statement: Let $X_0$ be scheme proper over separably closed field such that $dim X_0…
Marko markovictangens
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Properties of proper maps using filters

I am reading a book on covering maps in Bourbaki-style with the following definitions: A map $f:X\to Y$ is separated if for every $x,x'$ in the same fiber, there exist open disjoint neighbourhoods of $x$ and $x'$. A map $f:X\to Y$ is proper if every…
Dr. Heinz Doofenshmirtz
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Definition of proper map and compactification?

A proper map is a continuous map such that compact sets have compact preimage. For locally compact Hausdorff spaces, it seems that a proper map can be extended to a continuous map between one-point compactifications. Is this extension unique? If it…
Eric
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Let $f:X \to Y$ be a quotient map, $X$ Hausdorff. If $f$ is a proper function, then $Y$ is Hausdorff.

I've proven that $Y$ is $T_1$ using $f$ proper $\Rightarrow $ $f$ closed and since X is Hausdorff, X is $T_1$, then the unitary sets are closed in $X$. $f$ is quotient so it's surjective, then for all $y \in Y$ exists $x \in X$ that $f(x) = y…
Christian Coronel
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About proper functions and Fourier transforms

I was wondering if anyone knows where I can find examples of proper functions and their corresponding eigenvalues. This question arises from the fact that $e^{-\pi x^2}$ is a well-known proper function or autofunction of the Fourier transform…
Isidoro
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Action of a crystallographic group on $\mathbb{R}^n$

$G$ is called $n$-dimensional crystallographic group if it's a discrete subgroup of $\operatorname{Isom}(\mathbb{R}^n)$ acting on $\mathbb{R}^n$ with compact fundamental domain. An action of a topological group $G$ on a topological space $X$ is…
Giuseppe Scola
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