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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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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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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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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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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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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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50 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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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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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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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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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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Verify some Steps in Open Mapping Theorem from Royden

I have two questions about statements in Royden's proof of the Open Mapping Theorem. He works up to the following inclusion ($B_X$ and $B_Y$ are unit balls in $X$ and $Y$, resp): $$\overline{B_Y} \cap ...
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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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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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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
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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
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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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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
26 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 ...
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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 ...