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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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An isometric mapping $f:(X,d) \to (X, d)$ is an open mapping

In general, the result doesn't hold. For example $f: \mathbb{R} \to \mathbb{R}^2$ with $f(x)=(x,0)$ [Euclidian Norm]. Now my attempt to prove the restricted case goes as follows: Let $V \subset X$ ...
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Do continuous injections preserve open sets?

Do continuous injections preserve open sets? I'm pretty sure that's true in euclidean space. If we let the singleton sets of integers generate the topology of the domain, and then identity map it to ...
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Question about the proof open mapping theorem

Here $$ L = \{T(x) : x \in X \textrm{ and } \|x\| \leq 1\}. $$ While going through this proof, I don't understand the step in red, namely why $y-p \in \overline{L}$, could someone explain this? I'...
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Saturated subsets of quotient map.

From an example in Munkres Topology:$$\\$$ Consider the projection map $\pi_{1}: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ onto the first coordinate; it is continuous and surjective. It is also ...
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Does there exist a continuous, open, and surjective map from $f\colon\mathbb{R}^n\to\mathbb{R}^m$ for $m>n$?

My question is that from above. Here are my approaches so far: I know that there is no homeomorphism between an open set of $\mathbb R^n$ and an open set of $\mathbb R^m$. So if there is an open set ...
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To show a norm is finer than other norm

Let $V$ and $W$ be two Banach spaces and let $T \in L(V,W)$ be such that $R(T)$ is close and dim $N(T)< \infty$.Let |.| denote another norm in $V$ with $|x|\leq M||x||_V$ for all $x\in V$.Prove ...
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Show a closed bounded set in $\mathbb{R}^1$ maps to a

Let $B \subset [1,2]$. Define the set $E \subset C[1,2]$ by $$E = \{x \in C[1,2]: |x(t)| < 1 \text{ if } t\in B\}$$ Prove that if $B$ is closed in $[1,2]$, then $E$ is an open set in $C[1,2]$. I'...
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Motivation: Open Mapping Theorem

One of the Fundamental Theorems in Functional Analysis is the Open Mapping Theorem. Theorem. Let $X,Y$ be Banach spaces and $T \in L(X,Y)$. Then $T$ is surjective if and only if it is open. I'd love ...
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Show that $f$ is an open mapping (using Inverse Function Theorem)

This is an exercise for my complex analysis course. I have some ideas about this exercise, but I am not sure if I am correct. Use the Inverse Function Theorem to show that if $f: A \subset \mathbb{C} ...
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Every continuous open mapping from $\mathbb{R}$ into $\mathbb{R}$ is monotonic

Consider the image of an open set $(a,b)$ under the open and continuous mapping $f$. We show, $f$ cannot have any extremum in $(a,b)$. We know, connected sets are mapped to connected sets under a ...
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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 ...
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Compact open operator between Banach spaces

Let $X,Y$ be Banach space, $Y$ infinite dimensional. Show that no $T \in \mathcal{K}(X,Y)$ is open. By definition $T$ is open if and only if $\exists r >0$ such that $B_Y(0,r) \subset T(B_X(0,1))$ ...
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Application of the open mapping theorem on sequences

Let $X$ and $Y$ be Banach spaces and $A\in B(X,Y)$ surjective operator. I know that from open mapping theorem follow that there exist $C>0$ such that for every $y\in Y$ exist $x\in X$ such that $Ax=...
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44 views

Open Mapping Theorem fails when space is not complete

I am working on the open mapping theorem, and I want to show that completeness is a necessary condition for it to work. I have shown that if $X$ is Banach, $Y$ is a normed space, then there exists a ...
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completeness and the open mapping theorem

Let $X,Y$ be normed vector spaces, I want to show that the open mapping theorem requires completeness of both spaces. So my question consists of two parts: $\textit{i)}$ Let $X$ be a Banach space and ...
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Quotient maps and open maps

I was doing Exercise A.36 in Lee's Introduction to smooth manifolds which states the following: Let $q: X \rightarrow Y$ be an open quotient map. Then $Y$ is Hausdorff if and only if $R = \{(x_1, ...
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Linear continuous bijection but not open.

I have the next question. Let $l^1$ be the set of sequences $(a_1,a_2,\ldots, )$ such that $\sum |a_k|<\infty$. If we consider norm $|.|_1$ and the supremum norm $|.|_{s}$, then $(l^1,|.|_1)$ is ...
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Queris related to “An arbitrary product $X=\prod_{\alpha \in J} X_\alpha$ is connected iff $X_{\alpha }$ is connected for each $\alpha\in J$.”

The following proof of An arbitrary product $X=\prod_{\alpha \in J} X_\alpha$ is connected iff $X_{\alpha }$ is connected for each $\alpha\in J$ is provided by my professor.I've some ...
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Please check the proof of Imbedding theorem.

It is instructed in Munkre's topology that proof of imbedding theorem is almost the copy of step 1 of this post.It is instructed to just replace $n$ by $\alpha$ and $\mathbb R^{\omega}$ by $\mathbb R^...
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Spivak Differential Geometry 1 Chapter 1 Problem 8

Problem $8$ in Chapter $1$ of Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1 reads: 8. For this problem, assume (The Generalized Jordan Curve Theorem) If $A\subset \...
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Rudin functional analysis, theorem 2.11 (Open mapping theorem)

Going through the proof of such theorem, there are few bits I don't understand. I'll write down the proof and in in the middle I'll add some comments. Statement: Suppose (a) X is an F-space, ...
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1answer
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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 ) ...
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Quotient map from topological group onto left cosets is open

I've been trying to prove the following exercise from "James Munkers'-a first course in topology": Let $G$ be a topological group, and let $H\leq G$. Induce the left cosets, $G/H$, with the quotient ...
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Open mapping Theorem and Rouches Theorem

My question is related to the proof here: https://en.m.wikipedia.org/wiki/Open_mapping_theorem_(complex_analysis) Consider the closed Ball $B$ with radius $d$ around $z_0$ and a holomorphic function $...
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1answer
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How can the preimage of a closed set for an open map be open?

I am struggling to understand open maps. An open maps open sets to open sets. Given an open map between topological spaces $f : X \rightarrow Y$ If $U \in Y$ is open, $f^{-1}(U)$ can be open or ...
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1answer
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Is the box topology the finest that makes projections open maps?

There is a recurrent infinite/finite duality in topology, with one appearing in the opposite direction of the other; union/intersection, directed sets work their way upwards finitely while never ...
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1answer
53 views

Continuity property for a closed set

If $A$ and $B$ are topological spaces and $f:A\to B$ a continuous map and $U$ in $B$ a closed set, why is $f^{-1}(U)$ closed in $A$? I know that preimage of an open set needs to be open.
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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 ...
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1answer
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Difference between $T(A) = A$ and $T^{-1}(A) = A$

I am a bit confused about the title. Let $T:X\rightarrow X$ be a map where $X \subset \mathbb{R}^n$. Let $A\subset X$. I know that $$T^{-1}(A) = \{x\in X: T(x)\in A\}.$$ and if $A$ is $T$-...
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Exercise on quotient topology and countability axioms

Let $\, X := \mathbb{R}^3\Big/_{\sim} \,$ where $\, \sim \,$ is defined as: $\,x \sim y \iff x = y \quad \lor \quad \lVert x\rVert = \lVert y \rVert > 4$. Say wheter the canonical map $\, \pi :\...
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1answer
168 views

Is the inverse of a continuous function an open map?

The title is pretty self-explanatory, but I'll state the full question. Let $f: X \rightarrow Y$ be a continuous function between topological spaces. Is $f^{-1}$ an open map? By definition of ...
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Submmersions: $|f(x)|$ do not assume maximal value

I am studying analysis and I have had a lot of uncertainties. For instance, I cannot solve this exercise: If $f:U\rightarrow\mathbb{R}^3$ has class $C^1$ and rank $3$ in all of the points of the open ...
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Explicit/constructive example of open maps that are not continuous (especially from R to R)?

TLDR: I'm looking for an explicit map that is an open map but not continuous. The context my question arose was when learning the topological definition of continuous function. I made some progress ...
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Why does this statement hold?

I have seen the following statement: Let $G\subset \mathbb{R}^n$ be open, bounded and $f:\overline{G}\rightarrow \mathbb{R}^n$ a continuous and open map. Then $\|f\|$ gets its maximum on the ...
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If the closure of each $\mathcal{R}$-saturated set is also $\mathcal{R}$-saturated, is then $\mathcal{R}$ an open relation (projection is open)?

To give some background, if $(X,T)$ is a topological space, $C\subset X$, $\mathcal{R}$ is an equivalence relation in $X$ and $\pi:X\to X/{\mathcal R}$ is its canonical projection, then we call $C$ $\...
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The open $(0,1) \times (0,1)$ square invectively mapped *into* the interval $(0,1)$

Does the following bijection work: Take any point $(x,y) \in (0,1) \times (0,1).$ Each real number $r \in (0,1)$ may be represented by an infinitely-long decimal expansion (0.235, for example, is the ...
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2answers
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Problem based on Open Mapping Theorem in Functional Analysis

Let $V$ and $W$ be two Banach spaces and let $T\in L(V,W)$ be bounded such that $R(T)$ is closed and dim $N(T)<\infty$. Let $|.|$ denote another norm on $V$ with $|x|\leq M\|x\|_V$ for all $\,x\in ...
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How can I verify that this standard map is symplectic?

Show that the following standard map is symplectic $$ I(t+1) = I(t) + K\sin\theta(t) $$ $$ \theta(t+1) = \theta(t) + I(t+1) \bmod 2 \pi$$
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1answer
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Properties of inclusion map between topological spaces.

Let $X$ be a topological space and $Y$ a subset of $X$. Write $i: Y \to X$ for the inclusion map. Choose the correct statement: If $i$ is continuous, then $Y$ has the subspace topology. If $Y$ is an ...
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Map $F$ open and $F(\mathbb{R})$ is a submanifold

Let $F: \mathbb{R} \rightarrow S^1 \times S^1$ defined by $$F(t) = ((\cos2\pi t, \sin2 \pi t), (\cos2\pi \alpha t, \sin2 \pi \alpha t))$$. For which values ​​of $\alpha$ the map $F$ is open? For ...
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Function is open iff every element of a subbasis maps to open set?

I recently read that, given topological spaces $S,T$ and a map $f:S\rightarrow T$, for $f$ to be open it is sufficient to show that for a certain subbasis $C$ of $T$ and all (open) sets $A\in C$ holds ...
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1answer
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Showing the $2$-sphere with antipodal points identified is homeomorphic to the upper hemisphere with antipodal points identified.

Let $S_+$ be the closed upper hemisphere of the $2$-sphere $S^2$. We can define an equivalence relation $\sim_+$ on $S_+$ as follows: $x\sim_+ y\:\:\Leftrightarrow\:\:\begin{cases}x=x^\prime,&\...
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3answers
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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 ...
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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?
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1answer
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Canonical projection is open - Projective Space

Show that canonical projection $\pi:\mathbb{R}^{n+1}\setminus \{0\}\rightarrow \mathbb{RP}^{n}$ is open. I tried to prove it, but I have a hard time pulling the aberts. I search the internet but only ...
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1answer
101 views

Is a surjective continuous map with compact domain is open?

Let $f:X→Y$ be a continuous surjective map and $X$ is compact. Is $f$ is an open map? $"$A function $f : X → Y$ is open if for any open set $U$ in $X$, the image $f(U)$ is open in $Y$.$"$ Since $f$ ...
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1answer
196 views

Showing that a harmonic function maps open sets to open sets.

I am trying to show that a harmonic function maps open sets to open sets. I have written down a proof based on the hint provided by Theo Bendit here : Proof : Let $u : \Omega \to \Bbb R$ be a non-...
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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 ...
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0answers
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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 ...
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1answer
28 views

Bounded inverse (Brezis)

In the proof of the corollary 2.7 (T bounded, bijective so $T^{-1}$ bounded) the autor uses the conclusion of the open map theorem $TB_E(0,1)\supset B_F(0,c)$ and the injectivity of the Operator to ...