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.
Questions tagged [closed-map]
169 questions
112
votes
4 answers
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
- 1,471
39
votes
1 answer
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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27
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2 answers
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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21
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2 answers
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
- 503
18
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2 answers
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
- 12,914
16
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2 answers
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
- 161
15
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2 answers
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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14
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2 answers
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
- 1,930
12
votes
1 answer
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
- 129
10
votes
1 answer
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
- 87,968
10
votes
1 answer
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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9
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1 answer
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
- 27,542
7
votes
1 answer
Continuous surjection $\mathbb R^m\to \mathbb R^n$ that is not a quotient map
(I have found examples 1,2 that answer my original questions, so the question here is refined)
This question has been asked many times in this site, but all examples I see are maps between some complicated spaces. So here I'm asking for some…
Eric
- 1,194
7
votes
1 answer
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
- 2,897
6
votes
2 answers
A continuous function such that the inverse image of a bounded set is bounded
Suppose $f:\mathbb{R}\longrightarrow \mathbb{R}$ be an arbitrary continuous fuction such that the inverse image of a bounded set is bounded. Then show that,
$1$) The image under $f$ of a closed set is closed.
$2$) $f$ is not necessarily a surjective…
Shankhadeep
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