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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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Open maps which are not continuous

What is an example of an open map $(0,1) \to \mathbb{R}$ which is not continuous? Is it even possible for one to exist? What about in higher dimensions? The simplest example I've been able to think of ...
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Projection is an open map

Let $X$ and $Y$ be (any) topological spaces. Show that the projection $\pi_1$ : $X\times Y\to X$ is an open map.
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When quotient map is open?

Quotient map from $X$ to $Y$ is continuous and surjective with a property : $f^{-1}(U)$ is open in $X$ iff $U$ is open in $Y$. But when it is open map? What condition need?
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1answer
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Extending open maps to Stone-Čech compactifications

Let $X$ be a Čech-complete space, and $Y$ a paracompact space. Suppose $f\colon X\to Y$ is a continuous and open surjection. Since $Y$ is completely regular we have that $\beta(Y)$ is homeomorphic to ...
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Show that an open linear map between normed spaces is surjective.

Let $X,Y$ be normed spaces and $T:X\to Y$ is an open linear map. Show that $T$ is surjective. In order to show $T$ is surjective let's take $y_0\in Y$ and assume the contrary that $Tx\neq y_0\forall ...
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1answer
332 views

Attempted proof of an open mapping theorem for Lie groups

The Classical open mapping theorem for Banach spaces tells that if $T:X \to Y$ is a continuous surjective linear map, then it is open. I have attempted to essentially "adapt" the proof for Lie ...
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1answer
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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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1answer
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Opens maps from topological manifolds whose fibers are not generically topological foliations

Update. I have asked this on MO, but have not yet received an answer. Proposition. The quotient map associated to a topological foliation (projecting to the leaf space) is open. However the fibers ...
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Intuition for an open mapping

What is an intuitive picture of an open mapping? The definition of an open mapping (a function which maps open sets to open sets) is simple sounding, but it's really not as easy to picture as the ...
5
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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) ...
5
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1answer
269 views

Discontinuous surjective linear map which is not open

The following statement is true: Assume that $X$ and $Y$ are topological vector spaces where $Y$ is finite-dimensional Hausdorff, if $A:X\rightarrow Y$ is a continuous surjective linear map then $A$ ...
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Alternate topological definition of continuity

The standard topological definition of continuity is as follows: Definition: Continuity Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces. A function $f : X \to Y$ is ...
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2answers
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continuous image of a locally compact space is locally compact

Is continuous image of a locally compact space is locally compact? Let $X$ be locally compact(l.c.).Let $f:X\to Y$ is continuous and surjective. A space $X$ is locally compact if for each $x\in X$ ...
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1answer
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Is exponentiation open?

Already for $2\times 2$ matrices the exponential map is not open. However, the diagonalization trick does not work for algebras of functions. Hence the question Is the map $f\mapsto \exp(f)$ open on ...
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Let $Y$ be a finite dimensional normed space, $X$ a normed space, and $T: X \to Y$ a surjective linear operator. Show that $T$ is an open mapping.

Let $Y$ be a finite dimensional normed space, $X$ a normed space, and $T: X \to Y$ a surjective linear operator. Show that $T$ is an open mapping. I think if I can show that $T(B_X)$ contains an ...
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1answer
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Is convex-hull operation an open mapping?

Let $\mathcal{F}$ denote the set of all finite subsets of $\mathcal{R}^n$, endowed with Hausdorff metric, denoted by $d_H$. Let $\mathcal{C}=\{co(F):F\in\mathcal{F}\}$, also endowed with Hausdorff ...
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1answer
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If $X$ and $Y$ are Hausdorff, $X$ is compact, and $f:X\to Y$ is continuous and surjective, then $f$ is open.

I was given the following assertion: If $X$ and $Y$ are Hausdorff, $X$ is compact, and $f:X\to Y$ is continuous and surjective, then $f$ is open. However, I believe that I have a counterexample: ...
4
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1answer
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$f:\mathbb R^2 \to \mathbb R$ be a function , $|f(x)-f(y)|\ge 3\|x-y\| , \forall x,y \in \mathbb R^2$ ; is $f(\mathbb R^2)$ open in $\mathbb R$?

Let $f:\mathbb R^2 \to \mathbb R$ be a function such that $|f(x)-f(y)|\ge 3\|x-y\| , \forall x,y \in \mathbb R^2$ , then is it true that $f$ maps open sets of $\mathbb R^2$ to open sets of $\mathbb R$ ...
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1answer
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Openness of a map of $G$-spaces

This is question 10 from chapter 1 of Bredon, Introduction to Compact Transformation Groups. Let $G$ be a compact group, $X$ and $Y$ be $G$-spaces (Hausdorff spaces with a continuous $G$-action), and ...
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Closed Range Convolution Operator

Does there exists a nontrivial $f \in L^1(\mathbb{R})\cap L^\infty(\mathbb{R})$ such that $f\ast L^\infty(\mathbb{R})$ is a closed subspace of $L^\infty(\mathbb{R})$? I couldn't find any good ...
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2answers
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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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3answers
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Local compactness is preserved under continuous open onto mappings

If $f$ is a continuous open mapping of a locally compact space $(X,\tau)$ onto a topological space $(Y,\tau_1)$, show that $(Y,\tau_1)$ is locally compact. The definition of locally compact is that ...
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2answers
101 views

Show a non-constant, continuous function $f:\bar{D}\rightarrow \bar{D}$ is such that $f(\partial D)=\partial D$

Suppose $\bar{D}= \{z:|z|\leq 1\}$ and assume we have a non-constant, continuous function $f:\bar{D}\rightarrow \bar{D}$, that is holomorphic on the interior of $\bar{D}$ and such that $f(\partial D)\...
3
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1answer
86 views

Incorrect proof of the Open Mapping Theorem

I was following the proof of the Open Mapping Theorem in Lang's Real and Functional Analysis and something odd happened. I was able to simplify a lot his proof. Not only that, but I was able to ...
3
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1answer
723 views

Give example of $f$ that is open but neither closed not continuous (in 2D).

I'm trying to teach my self topology. The book I'm using has the following problem: Give an example of two subsets $X,Y \subseteq \mathbb R ^2$, both considered as topological spaces with their ...
3
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1answer
105 views

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 \...
3
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1answer
55 views

What is wrong with this proof of the Open Mapping Theorem?

I'm using the definitions from Rudins book Functional Analysis. Suppose $X,Y$ are topological vector spaces, $\Lambda: X \to Y$ is continuous and linear and $\Lambda(X)$ is of the second category in ...
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3answers
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easiest way to show the map to circle is open

What is the quickest way to show that $F:\mathbb{R}^{n+1}\setminus\{0\}\to S^n$, $x\mapsto \frac{x}{||x||}$ is an open map?
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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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0answers
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Image of locally compact space under a continuous open map

In Wayne Patty's book titled Foundations of Topology, local compactness is defined as follows: $X$ is locally compact at a point $p$ in $X$ provided that there is an open set $U$ and a compact set $K$ ...
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3answers
135 views

A non-continuous function which is open?

In this exercise $I = [-1, 1]$ and $\mathcal{E}_1$ is the euclidean topology. I was looking at the function: $f:(\mathbb{R},\mathcal{E}_1) \rightarrow (I,\mathcal{E}_1)$ defined by: $$ f(x) = \begin{...
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2answers
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Wanted: a function relating two metric spaces that is open but not continuous

It is known that the notion of a function from spaces $R$ to space $D$ being open (set $x$ open in $R$ implies image $f(x)$ is open in $D$) is independent from the notion of a function being ...
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1answer
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$g\circ f$ is open, $f$ continuous and surjective $\Rightarrow g$ is open

Let $(X, \tau_{1}), (Y, \tau_{2}), (Y,\tau_{3})$ be topological spaces. Let $f:(X, \tau_{1}) \rightarrow (Y, \tau_{2}), \ g:(Y, \tau_{2}) \rightarrow (Y,\tau_{3})$ be two functions. Show that: If $g\...
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2answers
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Prove that for an open and continuous map $f:X \to Y$ between topological spaces, it is $f(\tau_X)=\tau_Y$

Let $(X,\tau_x),(Y,\tau_Y)$ be topological spaces and let $f:X \to Y$ be a continuous and open function. Prove that $f(\tau_X)=\tau_Y$ This is an excercise of the problems set of my course on ...
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2answers
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If $f :{\Bbb R}^2 \to {\Bbb R}^2$ is a function satisfying $\| f(x) - f(y) \| \geq 3\|x-y\|$ , then $f$ is an open mapping and generalizing it

Let $f : {\Bbb R}^2 \to {\Bbb R}^2$ be a function such that $\| f(x) - f(y) \| \geq 3\|x-y\|$ ($\|\cdot\|$ denotes the usual metric in ${\Bbb R}^2$). Then I was trying to show that $f$ is an open ...
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3answers
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Trouble with Proving the One-Point Compactification of $(0,1)$

I am trying to prove that the one-point compactification of the interval $(0,1)$ with the Euclidean Topology is $S_{1} = \{(x,y)\in \mathbb{R}^{2}|x^{2}+y^{2}=1\}$ and I am having some trouble getting ...
2
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1answer
123 views

Why does an open map send generic points to generic point?

I might miss the forest for the trees but: The situation is (too) special, I think. Nevertheless I'll describe it: I have an open morphism $X\to Y$ of schemes (morphism is locally of finite type). ...
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1answer
337 views

Why is the quotient map from the sphere to the real projective plane open?

By 'open map' I mean that the quotient map $q\colon S^2 \to \mathbb{RP^2}$ maps open sets in $S^2$ to open sets in the projective plane $\mathbb{RP^2}$. My thinking is the following: $q\colon S^2 \...
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1answer
182 views

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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2answers
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Image of a closed set is closed under bounded linear transformation between Banach spaces?

Let $A,B$ be Banach spaces and let $T:A \to B$ be a surjective, bounded, linear operator. Let $A_1$ be a non-empty subset of $A$, then: $T(A_1)$ is closed if and only if $A_1+ \textrm{ker}(T)$ is ...
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1answer
169 views

Local homeomorphism iff discrete fibers and open and [?]

A continuous map does not tear. A local homeomorphism also "locally does not glue". This is captured in part by the fact it has discrete fibers: the points which are glued are isolated from each other ...
2
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1answer
217 views

Why is the stereographic projection a map from $S^2\to\mathbb{R}^2$?

How can I understand that the stereographic projection $$X=\cot\left(\frac{\theta}{2}\right)\cos\phi,\hspace{0.5cm}Y=\cot\left(\frac{\theta}{2}\right)\sin\phi\tag{1}$$ is a map from the surface of a ...
2
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1answer
102 views

Projection map for the product, uniform and box topologies

That the canonical projection map (from say $\mathbb{R}^\mathbb{N}$ to $\mathbb{R}$) is continuous for all three topologies is straightforward. But is it also open for the uniform and box topologies? (...
2
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1answer
125 views

Projection of topological space $X\times Y$ is an open map

I must prove that the projections $\pi_1:X\times Y\to X$ and $\pi_2:X\times Y\to Y$ are open, that is, they take open sets to open images. Since a basis element of $X\times Y$ is of the form $U\times ...
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1answer
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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'...
2
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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 ) ...
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2answers
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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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2answers
110 views

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 ...
2
votes
1answer
83 views

Solve the integral $\int_0^1 x^{\frac{1}{3}}(1-x)^{\frac{2}{3}}dx$

I would like to consider two ways to compute the (real) integral $\int_{0}^{1}x^{\frac{1}{3}}(1-x)^{\frac{2}{3}}dx$ using complex analysis: (i) By residues (ii) By Beta function My computations: (...
2
votes
2answers
28 views

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 $V \subset X$ ...