Questions tagged [open-map]

In topology, an open map is a function between two topological spaces which maps open sets to open sets. That is, a function f : X → Y is open if for any open set U in X, the image f(U) is open 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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51 views

How the author obtains $\|x\| < 1$ in this proof of open mapping theorem?

I'm reading the proof of open mapping theorem in Brezis' book of Functional Analysis. Let $T: E \to F$ be a linear continuous function between Banach spaces. We denote the unit open balls by $\mathbb ...
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30 views

Show that $\dim(Z) < \infty$ if $Z \subset T(X)$ for compact $T$

I'm uncertain if my reasoning of this proof is correct. What do you think? Problem Let $X,Y$ be Banach spaces and $T: X \to Y$ a compact linear map. Suppose $Z\subset Y$ is a closed subspace such that ...
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50 views

Norm equivalence in a Banach space [closed]

Let $E$ be a Banach space with two norm on it: $\|\cdot\|_1$ and $\|\cdot\|_2,\,$ such that $$\|x\|_1 \le \|x\|_2, \qquad\text{for all $x\in E$.} $$ Prove that the norms are equivalent. I was working ...
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31 views

Let $ \Omega _1, \Omega _2$ be domains and let $f:\Omega _1\rightarrow \Omega _2$ be biholomorphism. Is this $f$ an open function?

Let $ \Omega _1, \Omega _2$ be domains and let $f:\Omega _1\rightarrow \Omega _2$ be biholomorphism. Is this $f$ an open function? I want to show that $f$ is continuous, bijective, open to conclude ...
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How far is this view correct on strengthening and weakening of the topologies of the domain and the codomain?

Let $f:X \to Y$ be a map between infinite topological spaces. I want to know how correct is this view on how refining (strengthening) and coarsening (weakening) the topologies on both the domain and ...
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52 views

Open Mapping Theorem and the Bounded Inverse Theorem to show $T^{-1}$ is not bounded

Preface This is a problem in my Functional Analysis course. I'm really struggling to dissect part (b) any help/alternative solutions would be greatly appreciated on both parts of the question :) ...
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1answer
55 views

Is there such a mapping that is open mapping.

Let X be the banach space and Y be the normed space.$T\in \mathscr{B}(X,Y)$.The operator T is surjective.If Y is a banach space, then T is an open mapping from the open mapping theorem. My question is ...
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73 views

Does a homeomorphism between metric spaces map open balls to open balls?

A homeomorphism $h: (X,d_1) \rightarrow (Y,d_2)$ is a bijection between two metric spaces $(X,d_1)$ and $(Y,d_2)$ such that $h$ maps open balls of $X$ to the interior of open balls of $Y$ and $h^{-1}$ ...
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55 views

$T^*$ is surjective implies $T$ is bounded below. [closed]

Let $X,Y$ be banach spaces and $T: X \to Y$ a bounded linear map. I'm trying to show that if $T^*$ (adjoint of $T$) is surjective, then $T$ is bounded below, i.e. ${\exists} c>0$ s.t $||T(x)|| \ge ...
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45 views

A continuous open surjective map $f$ with compact fibers preserves Hausdorffness.

Let $f:X\to Y$ be an open, surjective, continuous map between two topological spaces $X$ and $Y$ with compact fibers. Let $X$ be a Hausdorff topological space. Then $Y$ is Hausdorff. I wanted to show ...
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For $f$ open and surjective, let $x \in X$ have a filter $F \to f(x)$. Then $\exists U$ a filter where $F \vdash f(U)$ and $f(U) \vdash F$.

Problem Statement: Let $f : X \to Y$ be an open surjective mapping. Let $x \in X$ have a filter $\mathcal{F}$ such that $\mathcal{F} \to f(x)$ in $Y$. Show that there exists a filter $\mathcal{U}$ in $...
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How to show this map is open?

Let $f$ be the canonical quotient map of $\mathbb R^{n+1} - \{0\}$ onto $\mathbb R P^n$. Restrict $f$ to the hyperplane $H$ away from zero, so let $g = f|H$. Then I want to show that $g$ is a ...
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31 views

What is the difference between an identification and an open identification?

I asked a similar question yesterday but studying my course further, I stumbled upon a definition of open identification that confused me. It says that open identification is an identification that is ...
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54 views

What is the difference between a continuous map and an identification map?

According to the definitions I have, the map in topology is continuous if the preimage of every open set is open. The map $f:X \to Y$ is called an identification map if it is continuous, surjective ...
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1answer
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Is $\varphi : \mathbb{R^2} -\{0\}\rightarrow S^1$ defined by $\varphi (x)=\frac{x}{|x|}$ a quotient map?

$\textbf{My problem:}$ Prove that $\varphi : \mathbb{R^2} -\{0\}\rightarrow S^1$ defined by $\varphi (x)=\frac{x}{|x|}$ is a quotient map. $\textbf{My attempt:}$ I need to prove that $V\subset S^1$ is ...
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1answer
109 views

Is there an open bijective map from $\mathbb{R}$ to $\mathbb{R}$ that is not continuous?

I came upon this when trying to solve a similar problem first: Open maps which are not continuous(1), which is essentially my problem without requiring the map to be bijective. To my knowledge, there ...
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1answer
117 views

Why can any homeomorphism on a topological space can be extended to its Čech-Stone compactification?

Let $X$ be a topological space. More times, I came across the statement that the Čech-Stone compactification (the "most general" compactification of a topological space) clearly has a ...
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2answers
87 views

Is the mapping $\mathbb{R}^2 \to \mathbb{R} ^2$ given by $(\xi_1,\xi_2) \to (\xi_1,0) $ an open mapping ?Yes/No

Is the mapping $f:\mathbb{R}^2 \to \mathbb{R} ^2$ given by $(\xi_1,\xi_2) \to (\xi_1,0) $ an open mapping ?Yes/No My attempt : yes ,I think it is open because collection of elements of the form $(\...
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47 views

On Banach-Schauder theorem (open-mapping)

Theorem Let $X$, $Y$ be Banach spaces and $T \in {\cal B}(X,Y)$ a surjective linear map. Then $T$ is an open map (that is, it sends open sets in open sets and the inverse $T^{-1}$ is continuous). Now,...
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An application of Open Mapping Theorem on the identity operator [duplicate]

I have an assignment in the Functional Analysis Course to submit which the reference book is Conway's Functional Analysis. I couldn't reach a solution nor did I find on the Internet. But I found that ...
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1answer
44 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 ...
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1answer
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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 ...
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1answer
70 views

Proving operator image is closed

Let $T:X\to Y$ be a linear bounded and surjective operator between Banach spaces $X$ and $Y$. I want to prove that if $A+\ker T$ is closed then $T(A)$ is closed. I tried using the open mapping theorem ...
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1answer
42 views

Proving this map to be open and that the quotient space is not Hausdorff

This question is from Section 5.1 ($T_0 - T_2$ spaces) of Wayne Patty's topology and I am struck on it. So, I am asking for help here. Let $I =[0,1]$ and define $x \sim y$ provided that $x-y$ is ...
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1answer
39 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)\}...
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1answer
95 views

largest open set such that $F(X) = (\operatorname{tr}(X),\det(X))$ is open

I need to find the largest open set in $M_2 (\mathbb{R})$ such that the function: $F:M_2 (\mathbb{R}) \rightarrow \mathbb{R}^2$ given by $F(X) = (\operatorname{tr}(X), \det(X))$ is open. I am lost ...
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56 views

Prove that function $g(x×y)=x+y^2$ is a open map [closed]

I have this definition: A map $f$ is open if the image of any open set $\mathcal{O}$ is open. I.e., if $f(\mathcal{O})$ is open. But I don't know how to use it, could you please guide me I thank you ...
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86 views

Continuous $f:$ open $\Omega \mapsto Y, Y$ (Banach) $\Rightarrow$ Inverse function is an open map

I'm self-studying Cheney's book in Functional Analysis, and this is among the exercises (problem 3.4.5). I think I've proven the claim in the problem, but I guess I've made errors along the way or ...
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2answers
128 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 ...
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1answer
49 views

What is an example of an open set A which we can write it in the form of a Cartesian product of two sets $B\times C$? (B and C not necessarily open) [closed]

I know each open set not necessarily be the cartesian product of two open sets. but I want an example for this equation. Please help me. Thank you in advance!
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1answer
29 views

Show the function is a open map

I was searching for a function which is open but not continuous and I got the following function: $f:[0,1]\to [0,1]$ \begin{equation} f(x)= \begin{cases} 2x& \text{$if\ 0\leq x \leq \frac{1}{2}$}\\...
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1answer
52 views

Homeomorphism of a Torus

I need to show that the function: $f(u,v)=((r \cos(u)+a)\cos(v), (r\cos(u)+a)\sin(v), r\sin(u))$, $f:U\to\mathbb{R^3}$, when $U=(0,2\pi)\times (0,2\pi)$ is an homeomorphism to the torus. I show that $...
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3answers
52 views

Function that is closed but not open and not continuous

So, I am trying to find a function with requirements of being closed but not open and not continuous which is defined on the real numbers and the usual topology. I was able to find many piecewise ...
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1answer
70 views

Show that $T(O)$ is not open in $Y$

How can I show that $T(O)$ is not open in $Y$? where we have that $O = [{x \in X : ||x||_X < 1}]$. $X:[ {x = (x_1, x_2, x_3, . . .) : x_n ∈ \mathbb{K}, \sum_{n=1}^{\infty}|x_n|< \infty}]$ and $||...
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1answer
52 views

A surjective linear mapping $L$ from $X$ to $Y$ is open iff there is a constant $c$ for each $y=Lx$ satisfying $\|x\| \leq c \|y\|$

So I have to show both directions. For the if direction, I began as below. Suppose such a $c$ exists for each pair of $x \in X$ and its corresponding $y = Lx \in Y$. I have to show that the image of ...
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1answer
44 views

$f$ Open, Closed and continuous but not a local homeomorphism [closed]

Give an example of topological spaces $X$ and $Y$ and a map $f: X \to Y$ that $f$ is an open, closed, and continuous but not a local homeomorphism. (The map and the topological spaces have to satisfy ...
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1answer
73 views

Prove that if $f$ is a continuous map which maps every open set to an open set then $f$ is monotonic.

My attempt:If the function is not monotonous then we can consider points $a < b$ and $b <c$ such that $f(a)<f(b)$ and $f(b)>f(c)$.My intuition behind this was to consider a subset $[a,c]$ ...
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1answer
24 views

How to prove the continuity of the following Homeomorphism?

Let $f$ be a function from the one-point compactification of $\mathbb{R}$ to the circle defined as follows $t\mapsto (x,y) = \begin{cases} \phantom{\lim\limits_{t\to\infty}} \left( \dfrac{1-t^2}{1+t^2}...
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1answer
75 views

about embedding or proper embedding

let $f:(0,1] \rightarrow \mathbb{R^2} , f(t)=\left(t \cos \left( \frac{1}{t}\right), t \sin \left(\frac{1}{t}\right)\right )$ is $f$ an immersion, embedding or proper embedding? $f$ is an immersion ...
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0answers
30 views

open map between smooth manifolds

Let $N, M$ be two smooth manifolds and $f: N \rightarrow M$ a smooth immersion. (i)-If $f$ is a homeomorphism on its image the $f(N)$. Then $f(N)$ is a submanifold. (ii)- If the dimension of $N, M$ ...
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1answer
49 views

show Banach Isomorphism theorem using Closed Graph theorem

Closed Graph Theorem: let $X$ and $Y$ be Banach spaces and let $T : X \to Y$ be a linear map with closed graph. Then $T$ is continuous. Definition: let $X$ and $Y$ be two normed spaces and let $T : X \...
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1answer
81 views

In Euclidean Neighborhood Retracts, the neighborhood have the $\mathbb{R}^n$ topology?

I have $X$ retract by retraction $r$ of its open neighboorhood $Y\in\mathbb{R}^n$. I'd like to write that, if $O\subset X$ is an open set, so $r^{-1}(O)$ is an open set in $\mathbb{R}^n$. But $r$ ...
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49 views

show Open Mapping Theorem using Banach Isomorphism Theorem

Open Mapping Theorem: let $X$ and $Y$ be Banach spaces and let $T : X \to Y$ be a linear surjective continuous map. Then $T$ is opened. Definition: let $X$ and $Y$ be two normed spaces and let $T : X \...
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28 views

Conjecture on openness of an analytic mapping

Consider a real analytic function $g: \mathbb{R}^m \rightarrow \mathfrak{M}_{\mathbb{R}}(n, k)$ to the set of $n \times k$ matrices where $k \ge n + 1$ such that $\forall x \in \mathbb{R}^m \quad \...
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0answers
25 views

Homemorphic spaces. [duplicate]

Let us consider $(X,\tau)$ a Hausdorff topological space y let $\infty$ an element such that $\infty \notin X$. Then $(\hat X, \hat \tau)$ is topological space, where $\hat X= X \cup \lbrace \infty \...
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0answers
66 views

Example of a bijective continuous map between Banach spaces whose inverse is not continuous

What would be an easy example of a bijective continuous map between Banach spaces whose inverse is not continuous? A usual example of a bijective continuous function between metric spaces would be $f(...
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4answers
91 views

What's wrong with my proof ??

This question was on my exam but apparently my answer was wrong as I only got half of the point: Suppose $p:X \to Y$ is an open map, prove if $A$ is open in $X$ then $q:A \to p(A)$ obtained by ...
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0answers
33 views

Which of the statements are true about the set $E \subseteq \Bbb R^3\ $?

Let $f : \Bbb R^3 \longrightarrow \Bbb R^3$ be given by $$f(x_1,x_2,x_3) = (e^{x_2} \cos x_1, e^{x_2} \sin x_1, 2x_1 - \cos x_3).$$ Consider a set $$\begin{align*} E & = \left \{(x_1,x_2,x_3) \in \...
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1answer
129 views

Open, discontinuous function from $\mathbb{R}$ to $\mathbb{R}$

I am looking for some examples of open functions from $\mathbb{R}$ (or $\mathbb{R}^n$) to $\mathbb{R}$ that are not continuous. I know that the classic example for an open, discontinuous function is ...
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2answers
37 views

$X$ is complete and $T$ is an onto open map, then $Y$ is complete

I am trying to prove sort of a converse to open mapping theorem. If $X, Y$ are normed linear spaces where $X$ is complete, and $T \in B(X, Y)$ is open, onto then I have to show Y is complete. I came ...

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