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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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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 a certain restriction of an open map open?

Let $p:X\rightarrow Y$ be an open map and let $A$ be a subspace of $X$. Then, is it true that $p|_A:A\rightarrow p(A)$ is open? I think so, but am struggling to show it. My thoughts: Let $O$ be an ...
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Showing $\{x\in X|f(\{x\}\times[0,1])\subseteq U\}$ is open

Let $f:X\times[0,1]\rightarrow Y$ be a continuous function, and let $U\subseteq Y$ be an open set. Show that the set $V=\{x\in X|f(\{x\}\times[0,1])\subseteq U\}$ is open in $X$. I don't really know ...
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Proving these equivalent conditions for an open map using boundary of a set

Let $X,Y$ be topological spaces. Prove the following statements are equivalent. $(1)$ $f\colon X\to Y$ is an open map. $(2)$ For all $x\in X$ and open set $U \ni x$ there exists open set $V$ ...
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Showing that the map $f(x)=x/norm(x)$ is an open map

I have to show that the map $f(x)=\dfrac{x}{\lVert x\rVert}$ is an open map where $x$ belongs to $\mathbb{R}^{n+1}/{0}$ and the range is $\mathcal{S}^n$ with the subspace topology. That this map is ...
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Preimage of a closed set under surjection

Let $X,Y$ be topological vector spaces, $\overline{B(0,1)}\subset Y$ is compact, and $T:X\to Y$ is a surjection. Would like to show that $W\subset Y$ is closed then $T^{-1}(W)\subset X$ is closed. My ...
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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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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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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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Analytic function being open

I was reading this question where it says that an analytic complex function $f:D \to \mathbb{C}$ are open mappings when $f′(z)$ is never zero. I´m not clear on two things, pardon me if they are ...
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Why is the inverse of a bounded bijective operator continuous?

I'm facing some troubles with the following theorem, Let $X,Y$ be Banach spaces and let $T \in B(X,Y)$. I want to show that if $T$ is bijective then its inverse is continuous. Now if $T$ is bijective ...
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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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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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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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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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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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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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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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Question About Proof of the Open Mapping Theorem

Below is a proof of the Open Mapping Theorem: I understand the proof, except I am unsure how we can guarantee that there exists a $\delta >0$ such that $f(z)\neq w_0$ for all $\delta = \mid(z-z_0)\...
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$f$ is open iff $f(U)$ is open for every $U$ open set of a fixed point $x \in X$ in a homogeneous space

Let $X$ be a homogeneous space (i.e. for every $x,y \in X$ there is a homeomorphism $\phi : X \to X$ such that $\phi(x)=y$) and $x \in X$ a fixed point, if $f$ is a function such that $f(U)$ is open ...
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How to think of an open ball in $\mathbb R^2$.

I'm having some difficulty understanding what exactly "open ball" means when it comes to dealing with subsets of $\mathbb R^2$, and likely by extension any metric space. Consider the subsets of $\...
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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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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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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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Showing $\forall q\in\mathcal{P}_Y$ there exists $ p\in\mathcal{P}_X:q\circ A\le p\implies A:X\to Y$ continuous

Let $(X,\mathcal{P}_X)$ and $(Y,\mathcal{P}_Y)$ be locally convex topological vector spaces with topologies induced by the families of continuous seminorms $\mathcal{P}_X$ and $\mathcal{P}_Y$ ...
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Is a faithfully flat map of affine schemes open?

$\newcommand{\Spec}{\mathrm{Spec}}$ Let $R \rightarrow S$ be a faithfully flat (unital) homomorphism of commutative rings. Does it follow that the corresponding map of topological spaces $\Spec S \...