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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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Best quantitative version of open mapping theorem in complex analysis

There is a "quantitative" version of the Open Mapping Theorem in complex analysis, saying: Let $f$ be holomorphic on the closure of a disc $V$ around $c$, and assume $m := \min_{z\in \delta ...
Torsten Schoeneberg's user avatar
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Let $f \in \mathbb{A}ut(\mathbb{D)}$. Show that $\lim_{|z|\to 1}|f(z)|=1$.

I have to following problem to solve: Let $f \in \mathbb{A}ut(\mathbb{D)}$. Show that $\lim_{|z|\to 1}|f(z)|=1$. (Without using the precise form of the automorphism.) So far I've got the following ...
Felix U's user avatar
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1 answer
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Orientation covering

My question is about whether or not the image of an open set is still an open set. I'm going to write the construction of the orientation covering and then I'll ask what I can't figure out. Let $M$ be ...
Gabriele's user avatar
1 vote
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$A:\mathbb{R}^n\to\mathbb{R}^m$ is a linear mapping. $V$ is an open set in $\mathbb{R}^n$. $A(V)$ is open in $A(\mathbb{R}^n)$. (Walter Rudin) [duplicate]

I am reading "Principles of Mathematical Analysis Third Edition" by Walter Rudin. The author says Since $V$ is open, it is clear that $A(V)$ is an open subset of its range $\mathcal{R}(A)=...
佐武五郎's user avatar
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The projection of an open ball is also an open ball

My problem is as follows: Let $\pi_i:\mathbb{R}^n\to\mathbb{R}$ be the projection onto the i-th coordinate. Prove that if $A \subset\mathbb{R}^n$ is open then its projection $\pi_i(A) \subset \mathbb{...
Kawanardo Queiroz's user avatar
10 votes
1 answer
370 views

If $F:\mathbb R^m\to \mathbb R^m$ is continuous with $|F(x)-F(y)|\geq \lambda|x-y|$, then $F$ is surjective.

I know that this question has been asked for several times (such as this post and the one-dimensional case). However, I still couldn't find an answer without using invariance of domain. The original ...
SuperSupao's user avatar
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Proving the set of surjective bounded linear operators is open in Banach space [duplicate]

Problem: Let $X$ and $Y$ be Banach spaces. Prove that the set $\Omega = \{T \in L(X,Y) \mid T \text{ is surjective}\}$ is an open set in the Banach space $L(X,Y)$ equipped with the operator norm. ...
Matrix AC's user avatar
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continuous open mapping of segment

$f : [0,1] \rightarrow [0,1]$ is a continuous open mapping. Show that it has finite number of maxima and $\max(f(x)) = 1$. It is well-known that if $g:\mathbb{R}\rightarrow\mathbb{R}$ is continuous ...
GeoArt's user avatar
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Non-surjective linear operator that is open?

Is there any non-surjective linear operator that is open? By the open mapping theorem, $T:X \to Y$ is open iff $T$ is surjective. Is it possible then to have a non-surjective linear operator that is ...
juan19.99's user avatar
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4 answers
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Prove a mapping is open

Prove that the map $$f:(0,1)\to\mathbb{R}^2$$ $$t\mapsto (\cos 2\pi t,\sin 2\pi t)$$ is an embedding. PS: In general topology, an embedding is a homeomorphism onto its image. More explicitly, an ...
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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
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2 votes
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Counterexample of Open Mapping Theorem

The point of the exercise is to show the necessity of requiring both normed spaces to be Banach for the open mapping theorem (which states that any surjective bounded linear operator $T: X\to Y$ ...
soggycornflakes's user avatar
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Open mapping theorem for affine maps between Choquet simplices

I was curious if an analogue of the open mapping theorem existed for affine maps between compact convex spaces. I'm interested in a question like the following: Suppose that $\mathcal{K}, \mathcal{L}$...
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Proving a natural map is open

This actually have been asked and answered here. I still have question for proving that the natural map $p$ is open and i can't comment since my reputation is not enough yet, so i make a new question. ...
Realm143's user avatar
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Is $\mathcal F: L^p(\Bbb R) \to L^q(\Bbb R)$ for $1 < p < 2$ surjective?

Consider the Fourier transform $\mathcal F: L^p(\Bbb R) \to L^q(\Bbb R)$ for $1 < p < 2$. Is this map surjective? I suspect it isn't; if one had to extrapolate the behavior on $L^1(\Bbb R)$ (see ...
stoic-santiago's user avatar
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Prob. 39, Chap. 7, in Schaum's GENERAL TOPOLOGY: The images of sets of a basis of the domain space under an onto open map ...

Let $(X, \mathscr{T})$ and $\left(Y, \mathscr{T}^*\right)$ be topological spaces, let the function $f \colon (X, \mathscr{T}) \longrightarrow \left(Y, \mathscr{T}^*\right)$ be open and onto, and let $\...
Saaqib Mahmood's user avatar
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0 answers
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evaluate integral with the kernel of Green function (fundamental solution of Poisson equation) of infinite domain using conformal map

I must find the following definite integral $$\phi(x,y) = \int_{0}^{\alpha} f(\theta)\frac{-1}{2\pi} \ln \sqrt{(x-\rho\cos\theta)^2+(y-\rho\sin\theta)^2} \rho d\theta$$ for a given $f(\theta)$ which ...
Hosein Javanmardi's user avatar
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Prove that every $C^1$ submersion is an open mapping.

I must not lie I am feeling very shaky in this subject, and I study mathematics as a hobby, so I have no way to check if my reasoning is correct. The statement of the question is exactly as in the ...
khalelbm's user avatar
3 votes
1 answer
115 views

Is multiplication by a scalar an open map in topological groups?

If n is any non-zero integer and G is a topological abelian group, under what conditions is the continuous homomorphism $g \mapsto ng$ open or weakly open? If this map happens to be open for all n and ...
Pedro Lourenço's user avatar
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1 answer
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Boundedness of Linear Operators on Banach Subspace with Different Norm

I had this exercise on a functional analysis exam but I was unable to solve point iii). I solved points i), ii), and iv). I solved i) with the open map theorem, ii) using i) and iv) as an instance of ...
BetaNab's user avatar
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Does there exist a continuous open map from the closed annulus to the closed disk?

In this MSE post A function $f:\mathbb{R}^2\to\mathbb{R}^2$ that is open and closed, but not continuous., user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") ...
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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
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If $T$ has a continuous inverse, then $R_T$ is closed [duplicate]

Let $T$ be a continuous (thus bounded) linear map from a Banach space $X$ to a Banach space $Y$. I want to prove that if $T$ has a continuous inverse, then $R_T$ is closed, where $R_T=T(X)$ is the ...
TOMILO87's user avatar
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Topology Exercise - Solution Verification

Hi everyone I've got this long topology exercise where I'm not completely sure about each one of my answers. Let $\mathbb{D}:= \{ (x,y) \in \mathbb{R}^2: x^2+y^2\leq 1 \}$ and $\tau$ be the family of ...
Turquoise Tilt's user avatar
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
1 vote
0 answers
98 views

Open mapping theorem for holomorphic functions from corresponding statement about harmonic functions

Let $f := u + iv : \mathbb{C} \to \mathbb{C}$ be a complex analytic (holomorphic) function. We know that $u, v : \mathbb{R}^2 \to \mathbb{R}$ are individually open maps. I was wondering whether one ...
SMS's user avatar
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1 vote
1 answer
43 views

Find comparable topologies on $X$ and $Y$ such that there exists $f:X⟶Y$ which is continuous or open with respect two but not the others.

Let be $\mathcal T_X$,$\mathcal T'_X$ and $\mathcal T''_X$ three topologies on a set $X$ such that $$ \tag{1}\mathcal T'_X\subseteq\mathcal T_X\subseteq\mathcal T''_X $$ and analogously let be $\...
Antonio Maria Di Mauro's user avatar
1 vote
1 answer
947 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
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0 answers
64 views

Why aren't all surjective continuous homomorphisms of topological groups open?

My apologies if this a naive question. Let $\phi:G\rightarrow H$ be a continuous surjective homomorphism of topological groups. Denote the kernel of $\phi$ by $N$, then: $$\begin{align} f:G\times N&...
Chris's user avatar
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1 vote
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Open mapping theorem for Frechet spaces

In Conway's Functional Analysis textbook, exercise-IV.2.8 asks to prove the open mapping theorem for Frechet spaces. In Conway's book, a Frechet space is not required to be locally convex. Let $X,Y$ ...
user760's user avatar
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1 vote
1 answer
232 views

$f$ is open iff the image of the unit ball contains a ball centered at 0

Disclaimer: I asked a similar question earlier but it was poorly phrased and not specific enough. Upon editing that question I realized it was too different compared to the original, so I have deleted ...
CBBAM's user avatar
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Topology generated by maps and subbase

Consider a family of maps $(f_t)_{t\in\mathbb{T}} : S\to S'$. I would like to define a topology making this family continuous. To do so, I consider a topology $\mathcal{U'}$ on $S'$ and I define the ...
G2MWF's user avatar
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2 votes
1 answer
355 views

Holomorphic functions are open mapping via inverse function theorem

I have been reading about the open mapping theorem for non constant holomorphic functions, all the proofs involve theorems of complex analysis (like Rouche, argument principle). What if we just ...
shark's user avatar
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2 votes
2 answers
111 views

A question about the proof of the open mapping theorem (2.11) in Walter Rudin's Functional Analysis

Can refer to page 65-66 of this pdf. I actually have two questions about the proof. The first is, in proving the first part of (2), why $\overline{\Lambda (V_2)}$ contains a neighborhood of $0$? I ...
Yongyi Yang's user avatar
2 votes
2 answers
87 views

How can I show that $D:X\rightarrow Y, f\mapsto f'$ is an open map?

Let $X=C^1[0,1]$ and $Y=C[0,1]$, endow both spaces with the maximum norm. Now define $$D:C^1[0,1]\rightarrow C[0,1], f\mapsto f'$$ I want to show that $D$ is an open map. My idea was the following: ...
user1294729's user avatar
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3 votes
2 answers
292 views

Continuous injective function that fails to map open sets to open sets [duplicate]

I'm studying metric spaces and have just proved that continuous maps preserve open sets under pre-image. The book I'm learning from says to beware of what this theorem does not say: that a continuous ...
jet's user avatar
  • 477
3 votes
1 answer
183 views

Does a bijection always generate a natural "induced" topology? [duplicate]

Consider a topological space $(X_1, \tau_1)$ and a set $X_2$. Suppose as well that there existed some bijection $h$ from $X_1$ to $X_2$ (and viceversa). Now suppose that we attempted to construct an &...
aghostinthefigures's user avatar
3 votes
0 answers
44 views

Continuous open image of locally compact space

Let $(X,\tau)$ be a topological space. A subspace is called precompact if its closure is compact. Now, local compactness Definition 1 (Munkres): Every point $x$ has an open neighbourhood $U_x$ ...
Rathindra N. Karmakar's user avatar
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1 answer
37 views

Continuos function- bounded function

Use the definition $\epsilon - \delta$ of continuity for proof that if the function $f: \mathbb{R}\longrightarrow \mathbb{R}$ is continuous in a, then $f$ is bounded in an open interval centered on a. ...
Stev's user avatar
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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
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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
0 answers
139 views

An application of open mapping theorem in linear algebra.

Open mapping theorem,which is taught in functional analysis,states that if $X$ and $Y$ are two Banach spaces and $T:X\to Y$ be a surjective bounded linear operator ,then the map $T$ is open.Now I ...
Kishalay Sarkar's user avatar
3 votes
1 answer
68 views

Is the expectation operator an open map?

Let $\Delta [0,1]$ denote the set of Borel probability measures on $[0,1]$. $\Delta [0,1]$ endowed with the Prokhorov metric is a metric space, as we know. My question is, does the expectation ...
Canine360's user avatar
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4 votes
2 answers
548 views

Is there a continuous open surjective map from the 2-sphere to the 2-torus?

Is there a continuous open surjective map from the 2-sphere $S^2$ to the 2-torus $S^1 \times S^1$? [Some thoughts: Since both spaces are compact, any continuous surjective map is a quotient map. ...
ccriscitiello'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
0 votes
1 answer
173 views

How to characterize open sets in the unit circle

At the end of the second chapter of Munkres' "Topology" there are some supplementary exercises concerning topological groups. One such group is $(S^1, \cdot)$. My problem is that the unit ...
Matteo Menghini's user avatar
2 votes
1 answer
43 views

Example of a function such that the property $ \forall U \in \tau, f^{-1}(f(U)) \in \tau$ doesn't hold.

What continuous function $f$, set and topology $\tau$ could serve as a counter-example of the following property : $ \forall U \in \tau, f^{-1}(f(U)) \in \tau$ ?
lvo0224's user avatar
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2 votes
0 answers
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Bounded surjective linear map from $L^p$ to $L^r$ with $r < p$

I am thinking about the following problem. Let $1 \leq r < p \leq \infty$, and consider the spaces $L^r([0,1])$ and $L^p([0,1])$. A routine application of Holder shows that $L^p([0,1]) \subset L^r([...
darthsid's user avatar
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1 answer
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Determine if analytic mappings from $\mathbb{C}\setminus \{0\}$ and $\mathbb{C}\setminus [0,\infty)$ to open unit disk exist and if so, find them...

This is a question from a previous complex analysis qualifying exam that I'm working through to study for my own upcoming exam. I'm not sure how to approach it. I've included my thoughts on it, but I'...
Serafina's user avatar
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2 votes
4 answers
96 views

$\forall \alpha\in\mathscr{A},\; X_\alpha\cong Y_{\varphi(\alpha)}\implies \prod_\alpha X_\alpha\cong \prod_\beta Y_\beta$ (Homeomorphisms).

The Unclear Claim I'm having trouble understanding the following statement from Topology by James Dugundji chapter IV: $\mathbf{2.6}$ Corollary. Let $\{X_\alpha\mid \alpha\in\mathscr{A}\}$ and $\{Y_\...
Choripán Con Pebre's user avatar

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