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.
24 questions
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Attribution for Metrization Theorem
Let $f:X \rightarrow Y$ be a continuous surjection with $X$ a compact metric space and $Y$ a Hausdorff space. Then $Y$ is metrizable.
Does this theorem have a name? Who first proved it? Just ...
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Degree of proper maps between Riemann Surfaces
I was studying some material on Riemann surfaces and came across the fact that for non-constant holomorphic maps between compact Riemann surfaces, the degree of such a map is constant. Moreover, I’ve ...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 $f:D \overset{k:1}{\longrightarrow} G$ ...
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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 ...
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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 ...
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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 ...
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Proper in one argument implies total properness?
Let $X$, $Y$, $Z$ be locally compact Hausdorff spaces. Furthermore, $X$ is compact. We have a continuous map
$$f : X \times Y \to Z$$
satisfying that for every fixed $x \in X$ the map
$$f_x\colon Y \...
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