Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit 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.

2
votes
1answer
29 views

Projection map is closed or not?

If we consider projection of $\Bbb R\times \Bbb R$ onto the $x$-axis(codomain is $\Bbb R\times \Bbb R$) the map is not closed as we know the image of a hyperbola ($xy=1$, closed) is not closed. In my ...
0
votes
1answer
40 views

Proof Verification: Show that if Y is compact, then the projection is a closed map

I searched the site for other proofs that may be similar to mine and couldn't find one. I was hoping someone could review my variation in particular for correctness. Thanks in advance. Problem: ...
1
vote
1answer
50 views

Restriction section of a sheaf to a closed set?

I am doing an exercise from Hartshorne (II Ex 6.2) on divisors and I have come across an abuse of notation that I am not entire sure how to interpret. Let $X \hookrightarrow \mathbb{P}_{k}^{n}$ be a ...
1
vote
1answer
59 views

Find continuous functions $f,g$ such that $g\circ f $ is closed and continuous but neither $g$ nor $f$ is closed map.

Find continuous functions $f,g$ such that $g\circ f $ is closed and continuous but neither $g$ nor $f$ is closed map. Find continuous functions $f,g$ such that $g\circ f $ is open and continuous ...
0
votes
1answer
16 views

Uniform map of R3 colour space onto a discrete number line

I'm thinking about theoretical colour palette formation. I have to start with a locus of $1000^3$ points in $R^3$, with each colour component axis being a discrete variable $0 \le r, g, b < 1000$. ...
0
votes
1answer
20 views

About Closed Graph Theorem for conjugate linear map

Let $X$ be a Banach space over $\mathbb{C}$ and $T \colon X \to X$ be a conjugate linear map. If the graph $G(T)= \{ (x,Tx) \mid x\in X \}$ of $T$ is closed in $X\times X$, is $T$ continuous?
0
votes
2answers
36 views

Show that given a proper map, for each closed set F its image f(F) is closed

For my math class, I have to provide the following proof: Given two metric spaces $(X,d)$ and $(Y,\rho)$, a continuous map $f: X \rightarrow Y$ is called proper if $f^{-1}(K) $ is compact for each ...
0
votes
1answer
19 views

coverage mapping at covering spaces

the professor at university asked us: is a coverage mapping like P from X to Y a closed mapping or not. Also; is p an open mapping? i could prove that P is an open mapping but for proving that P is ...
2
votes
1answer
43 views

Open and Closed maps in topology atan example

I am reading a basic introductory book on topology. It is written that a continuous map f from one topological space X to a second topological space Y is open ( closed ) if it maps open ( closed ) ...
2
votes
3answers
168 views

Example of quotient map from Munkres book

I was reading the notion of quotient map, topology and space but ran into the following example. In this example I have understood almost everything except one moment: How to prove rigorously that $p(...
0
votes
1answer
48 views

Does the distance map send closed sets to closed sets?

Let $(X,d)$ be a metric space and $x_0$ be an element in $X$. Define $f:X \to \mathbb R$ by $f(x)=d(x_0,x)$. Does this map send closed sets to closed sets? Further, let $C$ be a closed subset of $X$. ...
3
votes
1answer
42 views

Closedness of $\varphi\circ\pi$ and $\varphi$ implying $\pi$ to be closed

Let $X,Y$ and $Z$ be topological spaces, and consider the projections $\pi\colon X\times (Y\times Z)\to Y\times Z$ and $\varphi\colon Y\times Z\to Z$. 1) Is it true that if $\varphi$ and $\varphi\...
5
votes
1answer
157 views

Prob. 8, Sec. 26, in Munkres' TOPOLOGY, 2nd ed: A map into a compact Hausdorff 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 $...
0
votes
1answer
32 views

How to check thet the set is closed but not clopen

It is clear how to proof that the set $A$ is open. I just need to find some sequence which elements belong to $A$ while its limit does not. It is also clear that in order to show that the set $A$ is ...
1
vote
2answers
75 views

Example of a proper surjection to Hausdorff space that is not perfect?

Use the definitions: A continuous map is perfect if it is a closed surjection with compact fibers. A continuous map is proper if the inverse image of each compact set is compact. (In contrast to ...
0
votes
4answers
94 views

An embedding that is a closed map

Suppose $f:A \rightarrow B$, $g:B \rightarrow A$, such that $gf=1_A$. If $B$ is Hausdorff, then $f$ is a closed map. Any hints would be really appreciated.
3
votes
1answer
36 views

Is $\text{SL}(2,\mathbb{R}) \to \mathbb{R}^2 \setminus \{0\},\; A \mapsto Ae_1$ a closed map?

Question: Consider the map from the title $$f \colon \text{SL}(2,\mathbb{R}) \to \mathbb{R}^2 \setminus \{0\},\qquad A \mapsto A\begin{pmatrix}1 \\ 0 \end{pmatrix}.$$ Is it a closed map, i.e. does ...
0
votes
1answer
94 views

Does “unbounded” necessarily imply “goes to infinity”?

I am trying to understand this question How does continuity implies closeness by this guy @Logan. In the proof linked to in the question (linked again here) Theorem 1.2, it states Suppose to the ...
1
vote
3answers
42 views

Let $S$ be a metric space, $ f: S \to \Bbb R$ continuous. Define $Z(f) = \{p \in S : f(p) = 0 \}$. Prove $Z(f)$ is closed.

Let $S$ be a metric space, $ f: S \to \Bbb R$ continuous. Define $Z(f) = \{p \in S : f(p) = 0 \}$. Prove $Z(f)$ is closed. I've come up with a proof... I just would like to know if it is logical ...
0
votes
2answers
70 views

Linear image of a closed cone. Is it closed?

Today in my optimization class we needed to use that a certain set, which was the linear image of the closed cone $ S = \{ y \ge 0 \} \subset R^m$ (that is, evey component of $y$ is greater than $0$)$ ...
-1
votes
1answer
36 views

How to show that linear applications are closed and open [closed]

Let $f: R^n\mapsto R^m$ a linear application. Is $f$ closed? Why? Is it open? Why?
0
votes
1answer
47 views

Is this an open/ closed set? (Pre image, continuity question)

Consider the set $\Omega = \{(x,y,z) \in \mathbb{R}^3: \sqrt{x^2+y^2}\leq z< 1\} $ The question said to show that it is neither closed nor open, but I am getting an apparent contradiction. Define ...
1
vote
0answers
39 views

Proving a set is open using pre image and continuity of a function

I was tasked with determining whether the set {(x,y,z)$\in R^{3}: \sqrt{x^2+y^2}\leq z\leq 1$} is open or closed. I am wondering whether the following approach is valid/ formally okay. I am going to ...
0
votes
1answer
200 views

Is a bijective morphism between affine varieties an isomorphism?

I'm trying to prove the following: Let $X,Y$ affine varieties (both irreducible) and a morphism $f:X\to Y$. The pullback $f^*:A(Y)\to A(X)$ is surjective $\Leftrightarrow$ $f$ is injective and $f(X)...
1
vote
1answer
39 views

Proving that $P:\mathbb{C}\rightarrow\mathbb{R}$, defined by $P(z)=Re(z)$ is open but is not closed.

looking in a geometric way i can see that $P$ is open, $P$ works like a projection in the line, but i'm having some dificulties to show this with calculations. for the second part, i want to consider ...
0
votes
0answers
23 views

Contiunity and closedness of the map $F$ and $f.$

$X=C^1([0, 1])$ and $Y=C([0, 1])$, both with sup norm. Define $F: X \rightarrow Y$ by $F(x)=x+x'$, ($x'$ is sign of derivative) and $f(x)=x(1)+x'(1)$ for $x \in X$ Then which of the following is true?...
4
votes
3answers
111 views

Showing that a space is Hausdorff

Suppose $X$ is compact and Hausdorff and that $f:X \to Y$ is continuous, closed, and surjective. How can I show that $Y$ is Hausdorff?
6
votes
1answer
314 views

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 ...
6
votes
1answer
166 views

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. ...
1
vote
1answer
67 views

Continuous map on a dense subset which is not closed

I'm trying to prove that if $p:X\to Y$ is continuous, $D$ is a dense proper subset of $X$ and $p^{-1}(y)\cap D$ is compact for every $y\in Y$, then the restriction $\left.p\right|_D$ is not closed. I ...
1
vote
1answer
39 views

Show that $\omega : I \to \mathbb{S}^1$ defined by $\omega(x) = e^{2\pi ix}$ is a closed map.

Show that $\omega : I \to \mathbb{S}^1$ defined by $\omega(x) = e^{2\pi ix}$ is a closed map, where $I = [0, 1] \subseteq \mathbb{R}$. What I did so far was note that since $I$ is closed in $\mathbb{...
2
votes
1answer
86 views

action of a compact group is closed

I was reading a proof in invariant theory, they used the fact that the action of a compact group $G$ ; namely, $\phi: G \times M \rightarrow M$, is a closed map. I don't see this such obvious. I ...
0
votes
2answers
317 views

The inverse of projection function is a closed map?

I have this questione about the projection, be: $$\pi :X \times Y \to X$$ if we consider $ \pi^{-1}(x): x \to x \times Y $ I want to know if this is a closed map, is easy to see that $\pi$ is not, ...
1
vote
0answers
29 views

Is the set of subspaces generated by positive vectors a closed subset of the Grassmanian?

Consider the spaces $G=\{V = (v_1|\dots|v_k)\,:\,v_i \in\mathbb{R}^n,\,\mathrm{rank}(V) = k\}$ and $G_+=\{V = (v_1|\dots|v_k)\in G\,:\,v_i \in[0,\infty)^n\}$. If $Gr(k,n)$ denotes the real Grassmanian ...
4
votes
0answers
141 views

Not sure about one step in the proof that closed immersions are affine (Hartshorne 3.11b)

This question is about showing that if $i: Y \longrightarrow X$ is a closed immersion of schemes with $X = \text{Spec }A$ affine, then $Y$ is affine. This question has had a lot of attention on this ...
1
vote
1answer
777 views

Function is Closed?

While reading the optimization textbook, the below proposition given: Let $f: X \mapsto [-\infty, \infty]$ be a function. If $ \text{dom}(f)$ is closed and $f$ is lower semicontinuous at each x$\in \...
2
votes
1answer
107 views

Embedding onto adjunction space

Let X and Y be topological spaces. Let A $\subset$ X be a closed set and $f : A \to Y$ a continuous map. Prove that the canonical map $Y \to Y \cup _fX$ is an embedding onto a closed subspace. $Y \...
2
votes
0answers
81 views

A technical question on the category of metric spaces

Consider the category $\mathcal{M}$ whose objects are metric spaces, and maps are continuous injective maps and the subcategory $$^c\mathcal{M} \hookrightarrow \mathcal{M},$$ whose objects are metric ...
3
votes
0answers
343 views

Help proving that an finite morphism of schemes is a closed map

I am trying to prove that a finite morphism $f: X \longrightarrow Y$ of schemes is a closed map. That is, that the image of a closed set is closed in $Y$. I am a little unsure about the proof that I ...
2
votes
0answers
102 views

What is the most general setting in which the inverse image functor has a left adjoint?

Let $f: X \longrightarrow Y$ be a morphism of schemes. We have that the functors $$ f^{-1}: \text{Sh}(Y) \longrightarrow \text{Sh}(X) $$ and $$ f_{*}: \text{Sh}(X) \longrightarrow \text{Sh}(Y) $$ ...
1
vote
4answers
116 views

Given a continuous and closed map $f$ between two metric space $E, F$, prove that $f(\overline{A}) = \overline{f(A)}$ for all $A \subset E$

I already know of this question ($f$ is continuous and closed $\Longleftrightarrow \overline{f(E)} = f(\overline{E})$ for all $E \subseteq M$) which is quite the same as mine. But I don't understand ...
6
votes
1answer
486 views

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 ...
0
votes
1answer
155 views

When is a quotient map to an orbit space a closed map?

$\pi :X\to X/G $ where $G$ is a group acting on X and $X/G $ is the orbit space associated with this action. Does $\pi $ always take closed sets to closed sets? (or in other words is $\pi$ a closed ...
3
votes
2answers
194 views

Natural quotient map

Let $X$ be a Banach space and let $M $ be a closed subspace of $X $. When is the quotient map $Q :X \to X/M $ closed? I got somewhere that it happens iff$ M=X$ or $M=(0) $. But I couldn't ...
0
votes
0answers
37 views

Showing the continuity of a function based on the continuity of another function

I am having a bit of trouble seeing why the following $f_\epsilon$ is continuous. Be $M \subset \mathbb{R}^n $ bounded and closed and $f : M \to \mathbb{R}^k $ continuous, then for a $\epsilon > 0$...
1
vote
1answer
252 views

Lower semicontinuity and closedness equivalence.

I am not able to finalize the proof of this statement, and it seems something is always omitted in textbooks. Here there is what I got so far. (I'm trying to find the proof for a classical real-valued ...
3
votes
1answer
294 views

Example of a linear operator whose graph is not closed but it takes a closed set to a closed set

I want an example of a linear operator $T:X\to Y$, where $X$ and $Y$ are normed linear spaces, such that graph of $T$ is not closed but $T$ maps closed sets of $X$ to closed sets in $Y$. For ...
2
votes
1answer
58 views

$X$ is compactly generated iff every proper map from $Y$ to $X$ is closed

I was trying to prove the implication from right to left. I say that a space $X$ is compactly generated if $X$ is T2 and the collection $\mathcal{K}$ of the compact sets of $X$ is a fundamental cover (...
-2
votes
2answers
60 views

Set closed. Hahn Banach theorem. Banach limits

I have a question: why is this set $A$ closed? $$A=\{x-x' \mid x \in l_\infty \}$$ Where $l_\infty$ is the set of bounded sequences, $$x=( x (1),x (2), x (3),\dots),$$ and $$x'=(x (2),x (3),x (4),\...
5
votes
1answer
1k views

Open and Closed mapping Examples

I am looking for three mappings f:X to Y any set of topology on X or Y. so very flexible. Can you help me find an example of a function that is (a) continuous but not an open or closed mapping (b) ...