Skip to main content

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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
23 views

How can this open map characterization be explained or interpreted?

I hope you're having a good day. I'm an undergrad mathematics student, I took a general topology course a few months ago, and I'm now reviewing topological spaces to prepare for functional analysis. I ...
MHDFrags's user avatar
1 vote
1 answer
44 views

Surjective, closed local homeomorphism with finite fibers is a covering map?

Let $E,X$ be connected, locally path connected topological spaces and $\pi: E \to X$ a surjective, closed local homeomorphism with finite fibers. Is $\pi$ necessarily a covering map? I know that this ...
J. C.'s user avatar
  • 764
0 votes
0 answers
59 views

Determine which of these four sets are subspaces of the space $X$ and which of these subspaces are closed.

Let $X = C[-1, 1]$ be equipped with the usual maximum norm. Let $Y_1 = \{ f \in X \mid f(-1) = f(1) \}$, $Y_2 = \{ f \in X \mid f(-1) < f(1) - 1 \}$, $Y_3 = \{ f \in X \mid \int_{-1}^1 f(x) dx \...
user avatar
1 vote
0 answers
71 views

An example of an open map that is not closed

In Munkres Chapter 2.22 Example 1, he provided the following example of a continuous map that is closed but not open: Let $X$ be the subspace $[0, 1] \cup [2, 3]$ of $\mathbb{R}$, and let $Y$ be the ...
3m115's user avatar
  • 48
1 vote
1 answer
57 views

General topology question, homeomorphism to a closed subspace of $Y \times K$, $K$ is compact.

I have a quite interesting problem that I have to solve, and there're way too many conditions that I can't put all together. The problems is following: Let $X$ be a Tychonoff space (i.e. completely ...
Anton's user avatar
  • 383
1 vote
2 answers
216 views

If $Y$ is a metric space and $f: X \rightarrow Y$ is proper, $f$ is closed

Def: Let $f : X \rightarrow Y$ be a continuous map. $X,Y$ Topological spaces. $f$ is called proper if $f^{-1}(K)$ is compact for every compact $K \subseteq Y$. I want to prove that : If $Y$ is a ...
some_math_guy's user avatar
0 votes
1 answer
54 views

Is the function $f \colon (0, +\infty) \longrightarrow [-1, 1]$ defined by $f(x) = \sin (1/x)$ a closed map?

Let $(0, +\infty)$ and $[-1, 1]$ have the subspace topologies these sets inherit from the set $\mathbb{R}$ of real numbers with the usual topology. I know that the function $f \colon (0, +\infty) \...
Saaqib Mahmood's user avatar
0 votes
0 answers
83 views

How to show directly that the function $f \colon \mathbb{R} \longrightarrow \mathbb{R}$, $x \mapsto x^2$ is a closed map?

Let the function $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ be defined by $f(x) := x^2$ for all $x \in \mathbb{R}$, where the set $\mathbb{R}$ of real numbers has the usual topology. Then $f$ is ...
Saaqib Mahmood's user avatar
1 vote
1 answer
18 views

Mappings, Serge Lange Basic mathematics. (f.g)^3 = I

I have been struggling with this question for quite a time now: Let $f,g$ be mappings of a set $S$ into itself. Assume that $f^2 = g^2 = I$ and that $f \circ g = g \circ f$. Prove that $(f \circ g)^2=...
Zeeshan Amjad's user avatar
3 votes
0 answers
93 views

Does there exist a non-continuous clopen function $g: \mathbb R \to \mathbb R$? What about $\mathbb R^n\to \mathbb R^m$?

Inspired by Can an open and closed function be neither injective or surjective., but focusing on the case where $X,Y=\mathbb R$. First off, because the only nonempty clopen set in $\mathbb R$ is $\...
D.R.'s user avatar
  • 8,945
1 vote
1 answer
135 views

Let $(X, \mathcal{T})$, $(Y, \mathcal{U})$ be topological spaces, and let $f: X \rightarrow Y$ be a bijection. Then $f$ is open iff $f$ is closed

Here's my proof of the above statement. Since $f$ is open, it maps open sets in $\mathcal{T}$ to open sets in $\mathcal{U}$. Let $U \in \mathcal{T}$ be a closed set. So, $X \setminus U$ is an open set....
Ryukendo Dey's user avatar
0 votes
1 answer
48 views

A ball in $H^m(\Omega)$ is closed under $L^2$-convergence [closed]

I read it in a paper in which the author aims to claim that if a sequence $\{u_n\}$ is bounded in $H^m(\Omega)$ (here $\Omega$ is a bounded domain), and $u_n$ has a limit $u$ in $L^2$ norm, then $u$ ...
Jacob's user avatar
  • 11
0 votes
1 answer
178 views

Example of a closed map which is not continuous.

A map $F: X \to Y$ is said to be closed if $x_n \to x$ in $X$ and $F(x_n) \to y $ in $Y$ implies $y = F(x)$. As confusing as the definition is, I cannot seem to understand how a closed map is not ...
Snow's user avatar
  • 13
1 vote
3 answers
111 views

For a closed subset $F ⊂ X × Y$ , the image $π(F )$ need not be closed in Y

I have the following question from an exam If $(X, d_X)$ is compact, show that every sequence in $X$ has a subsequence converging to a point of $X$. Deduce that the projection map $\pi$ then has the ...
Maths Wizzard's user avatar
1 vote
1 answer
950 views

A linear transformation is open map if and only if surjective and closed map if and only if injective

Q.If $d$ and $e$ are positive integer and $T:R^{d} → R^{e}$ be a linear transformation then (a) $T$ is open map if and only if $T$ is surjective (b) $T$ is closed map if and only if $T$ is either zero ...
Sonu's user avatar
  • 563
0 votes
1 answer
66 views

Inverse image of a point under a continuous surjective closed map :

Let $f:X\rightarrow Y$ be a continuous surjective closed map and $X$ is a normal space. Let there exist an open set $U\subset X$ such that $f^{-1}\{ y \}\subset U$ then show that there exist an open ...
Infinity's user avatar
  • 645
1 vote
1 answer
328 views

Mapping a hemisphere onto the unit circle

How can I approach attempting to map a hemisphere onto the unit circle such that the meridians become arcs of a circle through $(0,1)$ and $(0,-1)$ and are evenly spaced around the x-axis (which is ...
Pen and Paper's user avatar
1 vote
1 answer
69 views

How can I show that $f\mapsto f'$ is a closed map? [duplicate]

Let $X=C^1([0,1])$ and $Y=C([0,1])$ both equipped with $\|f\|=\max |f|$. Then consider $$D:X\rightarrow Y:~~f\mapsto f'$$ I want to check that $D$ is a closed map. By definition we need to show that ...
user1294729's user avatar
  • 2,018
0 votes
1 answer
40 views

$S_1$: compact, $S_2$: any topological space. Then $S_1\times S_2\ni (x,y)\mapsto y\in S_2$ is a closed mapping.

I am reading "Introduction to Set Theory and Topology" (in Japanese) by Kazuo Matsuzaka. Definition: Let $(S,\mathcal{O})$ be a topological space. Let $x\in S$. Let $V\subset S$. If $x$ is ...
tchappy ha's user avatar
  • 8,750
1 vote
1 answer
54 views

Could you suggest a textbook for learning basic properties of spectrum of closed operators?

I wish to learn the very basic properties of the spectrum of Closed operators. I may use these properties in the research in the Fluid dynamics/Differential equations. My attempts:- I searched in ...
Unknown x's user avatar
  • 849
1 vote
0 answers
23 views

Example of closed map from $T_2$-space onto non-$T_2$ quotient? [duplicate]

What is an example of a continuous closed surjection $f : X \rightarrow Y$ from a $T_2$-space $X$ to a space $Y$ that is not $T_2$?
murray's user avatar
  • 780
0 votes
0 answers
85 views

$T-\lambda I$ is closed $\implies (T-\lambda I)^{-1}$ is closed

Let X be a complex Banach Space ,$T:X\rightarrow X$ a linear operator, and $\lambda\in \rho(T)$(Resolvent set),If T is assumed to be closed ,then I have to prove that $T-\lambda I$ is closed $\implies ...
Styles's user avatar
  • 3,569
9 votes
1 answer
260 views

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 $U \subset X$...
Paul Frost's user avatar
  • 78.2k
0 votes
1 answer
130 views

Characterization of Continuous, Closed and Open maps

I've been strugrilling trying to prove the following results from Lee's book on Topological Manifolds. Proposition 1. Let $f:X\to Y$ be a function between topological spaces. $f$ is continuous if ...
Nikolawn's user avatar
1 vote
1 answer
35 views

A question related to the closedness of a map between two topological spaces

Consider $\mathbb{R}$ with the Euclidean topology. Suppose we have an equivalence relation on $\mathbb{R}$ such that the equivalence classes are $\mathbb{Z}$ and single non-integer points. Let $q: \...
user avatar
2 votes
0 answers
63 views

$f: X \rightarrow Y$ injective, closed. Prove that for $U \in \mathcal{T}_x$ and $M$ ⊆ $Y$, ∃ $V \in \mathcal{T}_y$ s.t $M$⊆$V$ and $f^{-1}(V)$ ⊆ $U$

Let $(X,\mathcal{T}_x), (Y,\mathcal{T}_y)$ be topological spaces and $f: X \rightarrow Y$ be an injective and closed map (i.e. it maps closed sets to closed sets). Let $U$ be an open subset of $X$ and ...
Ph_Ys321's user avatar
1 vote
0 answers
55 views

Question about continuous but neither open nor closed function on topological space X

Define a function from topological space $\mathrm{X}$ into topological space $\mathrm{X}$, $f:\mathrm{X}\rightarrow \mathrm{X}$ such that $f$ is continuous but neither open nor closed My attempt I ...
beingmathematician's user avatar
1 vote
1 answer
42 views

The graph of a linear operator is closed but not continuous (closed graph theorem)

$\textbf{The question is}$ Let $X:=C^{1}[0,1] $, $ Y:=C^{0}[0,1]$ both equipped with the $\left \| f \right \|:=\begin{matrix} Sup & \left | f(x) \right |\\ x\in [0,1]& \end{matrix}$ and ...
F.R.'s user avatar
  • 178
0 votes
2 answers
39 views

Given domain $[0,T]$, can we have a 2nd order ODE describing $[0,T/2]$...?

If we have a whole time domain from $[0,T]$ can we have a second order ODE describing $[0,T/2]$ and then a first order ODE describing $[T/2,0]$ ensuring that the solutions of the ODE match up at $0$ ...
xatt1234's user avatar
-1 votes
1 answer
138 views

An example of a closed quotient map that is not open

We consider the map $$f\colon X:=[0,2\pi]\times [0,1]\to Y:=S^1\times [0,1],\quad\text{defined as}\quad f(t,s)=((\cos t, \sin t), s).$$ $f$ is a closed quotient map and corresponds to the equivalence ...
NatMath's user avatar
  • 162
3 votes
1 answer
286 views

How to prove T is a closable operator

Let $T:H\to H$ a densely defined operator, with $H$ a Hilbert space such that: $$Re(x,Tx)\geq 0, \forall x\in Dom(T) $$ I want to prove that $T$ is a closable operator, that means... that there exists ...
Blacks's user avatar
  • 403
1 vote
1 answer
82 views

Confusions about proof regarding proper maps

I am currently trying to prove that if $f:X\rightarrow Y$ is a closed continuous map between topological spaces such that $f^{-1}(y)$ is compact for all $y\in Y$ then f is proper. In this sense $f$ ...
Chris's user avatar
  • 3,431
0 votes
0 answers
45 views

Why is injectiveness needed to show that a bounded linear operator is a closed map?

Let $X$ and $Y$ be Banach. I'm trying to show that a bounded linear map $T:X\rightarrow Y$ is a closed map, that is $A\subset X$ is closed $\implies T(A)\subset Y$ is closed, if $T$ is injective and $...
stats19's user avatar
  • 103
0 votes
1 answer
63 views

An example of a non closed linear map.

Let $X,Y$ be Banach spaces. A map $T: X \to Y$ is said to be closed if: $$ x_n \to x \ \land \ Tx_n \to y \implies Tx = y$$ Do you have an example of a linear map that does not satisfy this ...
JustANoob's user avatar
  • 1,669
6 votes
2 answers
1k views

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 ...
Shankhadeep's user avatar
1 vote
2 answers
231 views

How does one conclude that a function maps closed sets to closed sets?

Let $f:(X,d_X)\to(Y,d_Y)$ be an equivalence (this means that $f$ is Lipschitz continuous, bijective and $f^{-1}$ is Lipschitz continuous). Let $K \subset X$. We want to prove that $K$ is closed and ...
Blue's user avatar
  • 293
0 votes
1 answer
48 views

$f:[0,2\pi]\rightarrow$ {$(x,y):x^2+y^2=1$} by $f(\theta)=(\cos\theta,\sin\theta)$ is a closed map

Define $f:[0,2\pi]\rightarrow$ {$(x,y):x^2+y^2=1$} by $f(\theta)=(\cos\theta,\sin\theta)$ I tried it by applying the definition of Closed map,but i'm unable to proceed. Please give some hint to ...
Styles's user avatar
  • 3,569
2 votes
1 answer
168 views

Pre-image of closed, surjective, continuous map with compact fibres is Lindelöf if its image is

I've been trying to prove the following result to no avail: Let $f: X \to Y$ be a continuous, closed, surjective map with compact fibres (i.e. a perfect map) between two topological spaces $X$ and $Y$....
Algebro1000's user avatar
2 votes
0 answers
124 views

Properness of a projection map is a local question, in a certain sense

I am studying the proof of Remmert's Theorem on the book Griffiths & Harris - Principles of Algebraic Geometry, Chapter 3 Section 2 page 395. There is this observation that I do not understand: In ...
PIELEO13's user avatar
0 votes
1 answer
2k views

Suppose $X$ is compact and $Y$ is Hausdorff show that $f$ is a closed map

I have seen a few other proofs about this. I have written my own, and I just want to verify the error in my own logic. The question states: Show that if $f:X\rightarrow Y$ is continuous, where $X$ is ...
Killaspe's user avatar
  • 338
3 votes
1 answer
425 views

Two definitions of a closed map

Def 1: Let $X$ and $Y$ be metric spaces and $T:X\to Y$, then $T$ is closed if $x_n \to x$ in X and $T(x_n) \to y$ in $Y$ implies $y=T(x)$. Def 2: Let $X$ and $Y$ be metric spaces and $T:X\to Y$, then $...
Sam's user avatar
  • 3,360
1 vote
1 answer
53 views

p closed continuous function $p(\bar A) =\overline {p(A)}$

Is it correct that when $p:X\rightarrow Y $ is a continuous and a closed map then if $A \subset X$: $$p(\bar A)= \overline{p(A)}$$ My attempt: As $p$ is a continuous function, then $p(\bar A) \...
Janbazif's user avatar
  • 433
3 votes
1 answer
83 views

Given a product topology X = $\prod_{i \in I}X_i$ , check: if E is closed in X, then $\pi_i(E)$ is closed in $X_i$

Given: A family of topological spaces: $\{(X_i, \mathcal{T_i})\mid i \in I\}$ $X = \prod_{i \in I}X_i$ is a product topology with topology defined as $\mathcal{T}$ $\pi_i: X \rightarrow X_i$ is ...
missmatchsox's user avatar
-1 votes
1 answer
42 views

Behaviour of ${f}^{-1}$ under continuous map

Let $f : \Bbb R \to \Bbb R$ be a continuous function. Which of the following is/are always true ? ${f}^{-1}(A)$ is open for all open sets $A \subseteq \Bbb R$ ${f}^{-1}(A)$ is closed for all closed ...
Mera bhai's user avatar
  • 301
0 votes
1 answer
69 views

Unable to prove that this map is a closed map.

This question was asked in my Topology quiz (now over) and I was unable to prove it Let $A=\{1/n : n \in \mathbb{N}\}$, and let $$B' = \{B\in \mathcal P(\mathbb{R}):(B\text{ is an open interval s.t. } ...
user avatar
0 votes
1 answer
108 views

How to prove that this map is not closed map

Consider following question: Let $X=\{(x,y) \in \mathbb{R}^2 : x\ge 0 \text{ or } y=0 \}$ and let $T$ be the subspace topology on $X$ induced by the usual topology on $\mathbb{R}^2$. Suppose $\mathbb{...
user avatar
1 vote
1 answer
46 views

closed quotient maps and $T_{k}$ spaces with $k \in \{1,2,3\}$

Let X be a topological space with $q: X \rightarrow X\setminus{\sim}$ the resulting quotient map. (i) q is a closed map $\implies$ for all $x \in X$ and for any open $U \subset X$ with $q^{-1}(\{q(x)\}...
Laura van Leuven's user avatar
1 vote
1 answer
76 views

Stacks Project, modules locally generated by sections; is the hypothesis necessary?

In the chapter Schemes of the Stacks project, I am confused about Lemma 4.5, which I state here. "Let $X$, $Y$ be locally ringed spaces, $\mathcal{I}\subset\mathcal{O}_X$ be a sheaf of ideals ...
Nico's user avatar
  • 1,543
0 votes
2 answers
962 views

If a continuous surjection $f$ is open/closed, then it is an identification.

Let $f: X \rightarrow Y$ be a continuous surjection between topological spaces. I have to show that: If f is open/closed, then it is an identification. Definition of an identification: A continuous ...
Laura van Leuven's user avatar
0 votes
1 answer
175 views

Banach fixed point theorem in $\mathbb R^n$

I have a question on Banach fixed-point theorem. It supposes we are in a closed set $C \subset \mathbb R^n$ with the image of $C$ by a function is included in $C : f(C)\subset C$ and we also suppose ...
Kilkik's user avatar
  • 1,952