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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Show the map $g:\mathbb{S}^1\to\mathbb{S}^1$ defined by $g(\cos(\theta), \sin(\theta)) = (\cos(a\theta), \sin(a\theta))$ is open where $a\in\Bbb N$.

For a function $g: \mathbb{S}^1 \to \mathbb{S}^1$ defined by $g(\cos(\theta), \sin(\theta)) = (\cos(a\theta), \sin(a\theta))$ where $a$ is an integer, how do I show $g$ is an open map? I know that the ...
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Proving a map is open

I have been trying to solve this "Problem 6." for days and have not made any improvements... Show that $f:X\to Y$ is open if and only if $f^{-1}[\operatorname{Fr}(B)]\subset\operatorname{Fr}[f^{-1}(...
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Showing $\mathbb{R} \setminus E = \{x \in \mathbb{R}: x \notin E\}$ is open.

Let $\mathbb{N}$ denote the set of all positive integers. In $(\mathbb{R}, d)$, show that $E = \{\frac{1}{n} : n \in \mathbb{N} \}$ is not closed, but $F = E \cup \{0\}$ is closed. A theorem in my ...
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Proving a function to be an open map

How can we prove that the function $f: S^1 \rightarrow S^1$ defined as $z \mapsto z^2$ is a continuous and open map using topological arguments? Here $S^1$ represents the unit circle in complex plane, ...
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Question in proof of open mapping theorem

Why is the mapping $\tilde A: X/N \to Y,\ \tilde A(x+N)=A(x)$ one to one (injective) and onto (surjective)? $A$ is a continuous, linear map from a F-space $X$ to a topological vector space $Y$ and $A(...
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Let X be a Banach space and show the series converges.

This problem comes from Royden & Fitzpatrick Real Analysis. Let $\{ u_n \}$ be a sequence in a Banach space $X$. Suppose that $\sum_{k=1}^{\infty} || u_k || < \infty$. Show that there is an $x ...
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Why is it sufficient to show that $f:X\rightarrow Y$ is open iff every image of an openset contains some nonempty openset

I can't wrap my head around the concept that to prove that $f:X\rightarrow Y$ is open, it is sufficient to show that every image of an openset $U\subset X$, $f(U)$ contains some nonempty openset $V\...
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Continuous $k : A \to B$ between two local homeomorphisms $f : A \to I, g : B \to I$ (i.e. such that $g \circ k = f$) is an open map.

It is known that every local homeomorphism $p : X \to Y$ is both continuous and open. Let $f : A \to I, g : B \to I$ be local homeomorphisms and $k : A \to B$ a continuous map be such that $g \circ k ...
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$p: \mathbb{R} \to \mathbb{R}/A$ Prove $p$ is an open map $\iff$ $A$ is open.

$A$ is a subset of $\mathbb{R}$ with more than 1 point and $p: \mathbb{R} \to \mathbb{R}/A$ is the quotient map. Prove that $p$ is an open map $\iff$ $A$ is open. I know if $A$ is open then for each ...
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Every local homeomorphism is an open map (Goldblatt's Topoi)

This is at the top of page 97. A sheaf is a bundle with some additional topological structure. Let $I$ be a topological space, with $\Theta$ its collection of open sets. A sheaf over $I$ is a ...
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$f: R^2 \rightarrow R$ defined by f(x,y) = x+y is open map but not closed map

Define $f: R^2 \rightarrow R$ by $f(x,y) = x+y$. I want to prove that 1. f is open, 2) f is not closed. My try: We can see that f is sum of the first and second projection maps which are open. But I ...
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Ball in closed convex set (from proof of open mapping thm)

(This is from the proof of the open mapping theorem, but this context isn't terribly important). $B(X)$ means unit ball in $X$. We have a bounded, linear, and surjective $T:X\to Y$ between Banach ...
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Techniques to Prove a Map Open or Closed

I've been finding it hard to verify that given maps are open or closed in practice. For example, in his discussion of the Mobius bundle, Lee's Introduction to Smooth Manifolds says without ...
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Fibre bundle which is not an open map [duplicate]

I am looking for a counterexample to the following claim: Let $p: E \rightarrow B$ be a fiber bundle, then $p$ is an open map. This is true when $E \simeq F \times B$ where $F$ is the fiber and $...
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A restriction of a closed map is a closed map?

$X,Y$ are topological spaces, $f:X \rightarrow Y $ is a closed map. Given a subset $B \subset Y$, let $A=f^{-1}(B) \subset X$. Prove that the restriction $g=f|_{A}:A \rightarrow B$ is a closed map. To ...
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Understanding the idea of the proof of a question.

Here is the question: Proving some properties of the identity operator. And here is the solution there: Let $U=\{f: \|f\|_{\infty} <1\}$. This is an open set in $(C[0,1],\|.\|_{\infty}\})$. Its ...
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Some questions regarding continuity and open and closed sets.

$f:X\to Y$ is a map between two topological spaces $X,Y$. $1.$ $f $ is said to be open (closed) mapping iff $f$ maps open(resp.closed) sets in $X$ to open sets in $Y$. $2.$ $f$ is said to be ...
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Use the Maximum modulus principle and the zeros of an analytic function.

Let $f$ and $g$ be analytic on a connected open set $U$. Assume that the closed disc $\overline{D(z_0,r)}$ is contained in $U$, where $r$ is a positive number. Show that if $|f(z)|=|g(z)|$ on the ...
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A question regarding the openness of a function on $\mathbb C^3$

$\mathbf {The \ Problem \ is}:$ Let $\phi : \mathbb C^3 \to \mathbb C^3$ be the map $(x,y,z) \mapsto (x+y+z,xy+yz+zx,xyz)$, then show that $\phi$ is an open map . $\mathbf {My \ approach} :$ First of ...
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Function Coefficients on Open Interval

Let $g:R^2 β†’R$ such that $g(x,y)=\sin y+y+e^x βˆ’1$ $\forall (x,y)∈R^2$ Prove function h exists such that it is defined on an open interval around the origin s.t. g(x,h(x)) = 0 for all x in ...
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When a quotient map of topological graph is open?

I follow the definition of a topology on a graph, from wikipedia: A graph is a topological space which arises from a usual graph $G=(E,V)$ by replacing vertices by points and each edge $e=xy\in E$ by ...
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Show that T is a surjective linear application

Let $X,Y$ be Banach spaces and $T:X\to Y$ a bounded linear application. Let $T^*:Y^*\to X^*$ be its adjoint (i.e. $T^*g(x)=g(Tx)$, $\forall g\in Y^*, x\in X$ ). If $||T^*g||\geq K||g||$ for some ...
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Minkowski sum of two sets in NLS [closed]

if $(X, \left\Vert \right\Vert)$ is an NLS, and U is open in X, then how to justify that $A+ U= \bigcup_{a\in A}(a+U)$ for any set $A \subsetneq X$
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Showing that a map is open by usying lower boundedness of adjoint

Let consider the following exercise: Let $V$ and $W$ be Banach spaces and $T\colon V \longrightarrow W$ be a closed linear operator. Then, the following assumptions are equivalent: (i) R$(T)=W$...
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Continuous open maps

If $f:X\rightarrow Y$ is a continuous open map then so is $f:X\rightarrow f(X)$ My attempt: Let $U$ be open in $f(X)$ so $U=U' \cap f(X)$ where $U'$ is an open subset of $Y$. Then $f^{-1}(U' \cap f(...
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Is there an open mapping theorem bewteen complete normed spaces without assuming linearity?

I read the following answer to a question where the OP clearly intended to use the open mapping theorem from functional analysis here. I'll add a more general version of the difficult direction, to ...
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Prove a real $f$ is injective if for all open sets $U$ in $\mathbb{R}$, $f(U)$ is open.

Question Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that for all open sets $U$ in $\mathbb{R}$, $f(U)$ is open. Prove that $f$ is injective. My proof Suppose $f$ is not ...
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What can I say about the function $f$? [closed]

Let $X= \left \{(x_i)_{i \geq 1} : x_i \in \{0,1 \}\ \text {for all}\ i \geq 1 \right \}$ with the metric $d \left ( (x_i),(y_i) \right ) = \sum\limits_{i \geq 1} \frac {|x_i-y_i|} {2^i}.$ Let $f: X \...
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“Net”-based definition of an open map [duplicate]

We know that the followings are among the two most important characterizations of continuity of functions between arbitrary topological spaces at a point. Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be two ...
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Is the canonical projection $p_i\colon\operatorname{lim}X\to X(i)$ an open map?

Let $X\colon I\to\mathsf{Top}$ be a functor, where $I$ is a category and $\mathsf{Top}$ is the category of topological spaces. We then can formulate the limit $\operatorname{lim}_IX$ (with a family of ...
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Openness of local real analytic map

Let $E,F$ - Banach spaces, $f: E \rightarrow F$ is a local real analytic isomorphism at every point in $E$. (analytic = continuously differentiable in Freshet sense, real analytic: let $E, F$ be ...
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“A continuous map which is open but not closed”-can someone explain to me why this proof works?

http://www.mathcounterexamples.net/continuous-maps-that-are-not-closed-or-not-open/ I need some explanation of a proof given in the above link. I'm looking at the proof that $f_1:(x,y)\longmapsto x$ ...
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Conditions for being an open map - check my proof

We define an open map as follows: Let be $f:M\to \mathbb{R}^n$, where $M \subseteq \mathbb{R}^n$. If $f(Q)$ is open for every open set $Q \subseteq M$ then we call $f$ an open map. Let's take an ...
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Proving $X$ is compact using a function $f\colon X \to Y$?

Let's suppose I have a topological space $X$, for which I am trying to prove its compactness. If I construct a function $f: X \to Y$ to another topological space $Y$, what are some examples of ...
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Sketch the open balls for the Railway Metric

Hi I'm having trouble picturing the open balls for the Railway metric 𝑑(π‘₯,𝑦)={𝑑2(π‘₯,𝑦) if π‘₯,𝑦,0 are collinear 𝑑2(π‘₯,0)+𝑑2(0,𝑦) otherwise I ...
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Homeomorphic Image of $G_{\delta}$ is $G_{\delta}$

Let $U=\cap_{n \in \mathbb{N}} U_n$ be a non-empty $G_{\delta}$-set; where each $U_n$ is open in a normal topological space $X$. If $\phi:X\rightarrow Y$ is a homeomorphism then is $\phi(U)$ a $G_{\...
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Questions about radially open sets topology. [duplicate]

I'm understanding this topology through doing an exercise. We say that a subset $U \subseteq \mathbb{R}^{2}$ is radially open if if for every $x \in U$ and every $u \in \Bbb R^2$ with $||u||=1$, ...
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Open mapping theorem calculus proof via Banach-Schauder theorem

Can the following theorem from calculus: Theorem: Let $F:\Omega\rightarrow \mathbb{R}^m$ be a continuously differentiable map, where $\Omega\subseteq \mathbb{R}^n$ is a domain(Connected and open)....
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Prove that $[0,\infty)$ is not homeomorphic to $\mathbb{R}$ without connectedness

I want to prove that the metric space $([0,\infty), |\cdot|)$ is not homeomorphic to $(\mathbb{R},|\cdot|)$ (or $((0,\infty),|\cdot|)$, whichever is easier) without using the notion of connectedness. ...
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Open map of varieties

Let $X_1$, $X_2$ be irreducible schemes locally of finite type over a field $k$. Let $f:X_1\rightarrow X_2$ be a bijective morphism of $k$-schemes. Can it map an open set to a non-open set? It can not ...
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1answer
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contious maps vs open maps in topological homeomorphisms

My lecturer writes in my topology notes that : A map $f:(X,\tau)\rightarrow (Y,\tau')$ is a homeomorphism if it is a bijection and both $f,f^{-1}$ are continuous. However he then goes on to write ...
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Continuous and one-one map is open if images of subbase open sets are open sets

In Willard's General Topology, in the proof of 8.12, page 56 (Dover edition), I find this: The aim is to prove that the map $e$ is an open map. $e$ is continuous and one-one as an hypothesis, and the ...
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let $f$ be a entire function and $B$ is bounded open set. $\partial(f(B)) \subset f(\partial B)$. [duplicate]

let $f$ be a entire function and $B$ is bounded open set. Then is it true that $\partial(f(B)) \subset f(\partial B)$. Hint is given that use open mapping theorem . But i got stuck how to use it. ...
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Proving a set is open in context of proof of a step in local Inverse Function Theorem

The context for this question is my answer to this question on the inverse function theorem. I'll try to replicate as much of the necessary information as possible: Let $E$ and $F$ be real Banach ...
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1answer
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Does every linear injective open map between Banach spaces map closed sets to closed sets?

Let $X$ and $Y$ be Banach spaces and $T \in L(X,Y)$, i.e. $T$ is a linear continuous map from $X$ to $Y$. Further let $T$ be injective and open. Does $T$ map closed sets to closed sets? I know this ...
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Is a differentiable (but not $C^1$) function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with invertible derivative everywhere an open map?

Is a differentiable (but not $C^1$) function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with invertible derivative everywhere an open map? I know that if we assume the function is $C^1$, then this is ...
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1answer
39 views

Want to show an embedding is open if the image is open

Let f be an embedding from X to Y. I want to show that if the image of X is open, then f is open. I know that a continious function is open if images of open subsets are open. Futheremore i know that ...
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40 views

Check whether the map $f(z)={z\over e^z-1}$ is open or not in $\{z|$Im$(z)>0\}$

As per open mapping theorem, a holomorphic, non-constant function on a region is an open map. Here the domain $\Omega=\{z\in\Bbb{C}|\operatorname{Im}(z)>0\}$ is a region, $f$ is non-constant. Here ...
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57 views

The converse of continuous image of a connected set is connected [duplicate]

I have recently proven that given $f:X\rightarrow Y, f$ continuous, X connected, then Y is connected. I wonder if the converse is true if we consider open mapping. In other words, Given X and Y ...
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2answers
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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 $V \subset X$ ...