Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [closed-map]

In topology, a closed map is a function between two topological spaces which maps closed sets to closed sets. That is, a function f : X → Y is closed if for any closed set U in X, the image f(U) is closed in Y. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.

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Projection map being a closed map

Let $\pi: X \times Y \to X$ be the projection map where $Y$ is compact. Prove that $\pi$ is a closed map. First I would like to see a proof of this claim. I want to know that here why compactness is necessary or do we have any other weaker…
M.Subramani
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Show that a proper continuous map from $X$ to locally compact $Y$ is closed

Let $f: X \to Y$ be continuous and proper (a map is proper iff the preimage of a compact set is compact). Furthermore, assume that $Y$ is locally compact and Hausdorff (there are various ways of defining local compactness in Hausdorff spaces, but…
JZS
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When is the image of a proper map closed?

A map is called proper if the pre-image of a compact set is again compact. In the Differential Forms in Algebraic Topology by Bott and Tu, they remark that the image of a proper map $f: \mathbb{R}^n \to \mathbb R^m$ is closed, adding the comment…
s.harp
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Are compact spaces characterized by "closed maps to Hausdorff spaces"?

It is well known that any continuous map from a compact space to a Hausdorff space must be a closed map. Does this fact characterize compactness? That is, if for a space $X$, every continuous map to any Hausdorff space is closed, does it imply that…
ronno
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Image of a normal space under a closed and continuous map is normal

$p : X \to Y$ is continuous, closed and surjective, and $X$ is a normal space. Show $Y$ is normal. There is a hint, which I'm trying to prove: show that if $U$ is open in $X$ and $p^{-1}(\{y\}) \subset U$, $y \in Y$, then there is a neighbourhood…
fuente
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A bijective continuous map is a homeomorphism iff it is open, or equivalently, iff it is closed.

Wikipedia states that "a bijective continuous map is a homeomorphism if and only if it is open, or equivalently, if and only if it is closed.". How do we prove this fact? I can prove the obvious direction, but im unsure how to proceed the other ways
terry
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Finite Product of Closed Maps Need Not Be Closed

What is an example of a finite product of closed maps that is not itself a closed map?
Rick
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Is projection of a closed set $F\subseteq X\times Y$ always closed?

If we have closed subset $F$ of product $X \times Y$ (product topology) does it mean that $p_1(F)$ (projection on first coordinate) is closed in $X$ and $p_2(F)$ in $Y$ are closed? If not, why not (some counterexample or explanation please =)
Meow
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Help understanding closed subschemes and closed immersions

Closed subschemes and closed immersions of schemes have been causing me a lot of confusion for a while now. I have a few questions that I think might clear things up. Please assume that when I use the term "ring" that I mean "commutative ring with…
Joe
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Continuous function from a compact space to a Hausdorff space is a closed function

We have that $f\colon X\to Y$ is a continuous function, $X$ is a compact space and $Y$ is a Hausdorff space. Prove that $f$ is a closed function.
Sokkun Yoem
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Prob. 8, Sec. 26, in Munkres' TOPOLOGY, 2nd ed: A map into a compact Hausdorff space is continuous iff its graph is closed

Here is Prob. 8, Sec. 26, in the book Topology by James R. Munkres, 2nd edition: Theorem. Let $f \colon X \to Y$; let $Y$ be compact Hausdorff. Then $f$ is continuous if and only if the graph of $f$, $$ G_f = \{ \ x \times f(x) \ \vert \ x \in…
Saaqib Mahmood
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Characterizing continuous, open and closed maps via interior and closure operators

A function $f :X \to Y$ between topological spaces $X,Y$ is defined to be continuous if $f^{-1}(V)$ is open in $X$ for all open $V \subset Y$, open if $f(U)$ is open in $Y$ for all open $U \subset X$, closed if $f(C)$ is closed in $Y$ for all…
Paul Frost
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A continuous surjection is proper if and only if pre-images of compact sets are compact

Dugundji's Definition: A map $f:X\to Y$ between topological spaces is called perfect (or proper), if it is a closed continuous surjection such that each fiber $f^{-1}(\{y\})$, $y\in Y$, is compact. Show that a continuous surjection $f:X\to Y$…
Derso
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Techniques to Prove a Map Open or Closed

I've been finding it hard to verify that given maps are open or closed in practice. For example, in his discussion of the Mobius bundle, Lee's Introduction to Smooth Manifolds says without justification that if $E$ is the quotient of $\mathbb{R}^2$…
Nick A.
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Articles on the "Property I found" and other types of Centers (excluding the Centroid)?

I am a first-year undergraduate student. I came up with a kind of center property which I cannot find in articles online. I found the center on my own but needed help of mathematicians (@Rahul) on this site. Currently I'm unsure whether it applies…
Arbuja
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