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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How to obtain spatial density from density?
Assume that $p=\chi^{-1}(x,y,z,t)$ is an invertible real-valued function, where $x,y$ and $z$ are real scalars and $t$ is time. Also assume that the inverse of $p$ is a vector-valued function, $(x,y,z)...
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Prob. 39, Chap. 7, in Schaum's GENERAL TOPOLOGY: The images of sets of a basis of the domain space under an onto open map ...
Let $(X, \mathscr{T})$ and $\left(Y, \mathscr{T}^*\right)$ be topological spaces, let the function $f \colon (X, \mathscr{T}) \longrightarrow \left(Y, \mathscr{T}^*\right)$ be open and onto, and let $\...
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evaluate integral with the kernel of Green function (fundamental solution of Poisson equation) of infinite domain using conformal map
I must find the following definite integral
$$\phi(x,y) = \int_{0}^{\alpha} f(\theta)\frac{-1}{2\pi} \ln \sqrt{(x-\rho\cos\theta)^2+(y-\rho\sin\theta)^2} \rho d\theta$$
for a given $f(\theta)$ which ...
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Prove that every $C^1$ submersion is an open mapping.
I must not lie I am feeling very shaky in this subject, and I study mathematics as a hobby, so I have no way to check if my reasoning is correct. The statement of the question is exactly as in the ...
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Is multiplication by a scalar an open map in topological groups?
If n is any non-zero integer and G is a topological abelian group, under what conditions is the continuous homomorphism $g \mapsto ng$ open or weakly open?
If this map happens to be open for all n and ...
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Boundedness of Linear Operators on Banach Subspace with Different Norm
I had this exercise on a functional analysis exam but I was unable to solve point iii).
I solved points i), ii), and iv). I solved i) with the open map theorem, ii) using i) and iv) as an instance of ...
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Does there exist a continuous open map from the closed annulus to the closed disk?
In this MSE post A function $f:\mathbb{R}^2\to\mathbb{R}^2$ that is open and closed, but not continuous., user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") ...
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Does there exist a non-continuous clopen function $g: \mathbb R \to \mathbb R$? What about $\mathbb R^n\to \mathbb R^m$?
Inspired by Can an open and closed function be neither injective or surjective., but focusing on the case where $X,Y=\mathbb R$. First off, because the only nonempty clopen set in $\mathbb R$ is $\...
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If $T$ has a continuous inverse, then $R_T$ is closed [duplicate]
Let $T$ be a continuous (thus bounded) linear map from a Banach space $X$ to a Banach space $Y$. I want to prove that if $T$ has a continuous inverse, then $R_T$ is closed, where $R_T=T(X)$ is the ...
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Topology Exercise - Solution Verification
Hi everyone I've got this long topology exercise where I'm not completely sure about each one of my answers.
Let $\mathbb{D}:= \{ (x,y) \in \mathbb{R}^2: x^2+y^2\leq 1 \}$ and $\tau$ be the family of ...
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Let $(X, \mathcal{T})$, $(Y, \mathcal{U})$ be topological spaces, and let $f: X \rightarrow Y$ be a bijection. Then $f$ is open iff $f$ is closed
Here's my proof of the above statement.
Since $f$ is open, it maps open sets in $\mathcal{T}$ to open sets in $\mathcal{U}$. Let $U \in \mathcal{T}$ be a closed set. So, $X \setminus U$ is an open set....
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Open mapping theorem for holomorphic functions from corresponding statement about harmonic functions
Let $f := u + iv : \mathbb{C} \to \mathbb{C}$ be a complex analytic (holomorphic) function. We know that $u, v : \mathbb{R}^2 \to \mathbb{R}$ are individually open maps. I was wondering whether one ...
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Find comparable topologies on $X$ and $Y$ such that there exists $f:X⟶Y$ which is continuous or open with respect two but not the others.
Let be $\mathcal T_X$,$\mathcal T'_X$ and $\mathcal T''_X$ three topologies on a set $X$ such that
$$
\tag{1}\mathcal T'_X\subseteq\mathcal T_X\subseteq\mathcal T''_X
$$
and analogously let be $\...
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A linear transformation is open map if and only if surjective and closed map if and only if injective
Q.If $d$ and $e$ are positive integer and $T:R^{d} → R^{e}$ be a linear transformation then
(a) $T$ is open map if and only if $T$ is surjective
(b) $T$ is closed map if and only if $T$ is either zero ...
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Why aren't all surjective continuous homomorphisms of topological groups open?
My apologies if this a naive question.
Let $\phi:G\rightarrow H$ be a continuous surjective homomorphism of topological groups. Denote the kernel of $\phi$ by $N$, then:
$$\begin{align}
f:G\times N&...
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Open mapping theorem for Frechet spaces
In Conway's Functional Analysis textbook, exercise-IV.2.8 asks to prove the open mapping theorem for Frechet spaces. In Conway's book, a Frechet space is not required to be locally convex.
Let $X,Y$ ...
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$f$ is open iff the image of the unit ball contains a ball centered at 0
Disclaimer: I asked a similar question earlier but it was poorly phrased and not specific enough. Upon editing that question I realized it was too different compared to the original, so I have deleted ...
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Topology generated by maps and subbase
Consider a family of maps $(f_t)_{t\in\mathbb{T}} : S\to S'$. I would like to define a topology making this family continuous. To do so, I consider a topology $\mathcal{U'}$ on $S'$ and I define the ...
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Unramified maps & separated maps, algebraic & geometric multiplicity
Let $f : X \rightarrow Y$ be a finite morphism of schemes with $Y$ connected. I'm interested these conditions:
$f : X \rightarrow X \times_Y X$ is open (unramified).
$f : X \rightarrow X \times_Y X$ ...
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Holomorphic functions are open mapping via inverse function theorem
I have been reading about the open mapping theorem for non constant holomorphic functions, all the proofs involve theorems of complex analysis (like Rouche, argument principle). What if we just ...
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A question about the proof of the open mapping theorem (2.11) in Walter Rudin's Functional Analysis
Can refer to page 65-66 of this pdf.
I actually have two questions about the proof. The first is, in proving the first part of (2), why $\overline{\Lambda (V_2)}$ contains a neighborhood of $0$? I ...
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How can I show that $D:X\rightarrow Y, f\mapsto f'$ is an open map?
Let $X=C^1[0,1]$ and $Y=C[0,1]$, endow both spaces with the maximum norm. Now define $$D:C^1[0,1]\rightarrow C[0,1], f\mapsto f'$$ I want to show that $D$ is an open map. My idea was the following:
...
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Continuous injective function that fails to map open sets to open sets [duplicate]
I'm studying metric spaces and have just proved that continuous maps preserve open sets under pre-image. The book I'm learning from says to beware of what this theorem does not say: that a continuous ...
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Proof that every continuous linear bijection between Banach spaces is a homeomorphism without using the open mapping theorem
My question is exactly what is in the title. I'm studying functional analysis and I have an idea for a proof of the open mapping theorem using quotients. The thing is, for it to work, I need to prove ...
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Does a bijection always generate a natural "induced" topology? [duplicate]
Consider a topological space $(X_1, \tau_1)$ and a set $X_2$. Suppose as well that there existed some bijection $h$ from $X_1$ to $X_2$ (and viceversa).
Now suppose that we attempted to construct an &...
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Continuous open image of locally compact space
Let $(X,\tau)$ be a topological space. A subspace is called precompact if its closure is compact. Now, local compactness
Definition 1 (Munkres): Every point $x$ has an open neighbourhood $U_x$ ...
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Continuos function- bounded function
Use the definition $\epsilon - \delta$ of continuity for proof that if the function $f: \mathbb{R}\longrightarrow \mathbb{R}$ is continuous in a, then $f$ is bounded in an open interval centered on a.
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Characterizing continuous, open and closed maps via interior and closure operators
A function $f :X \to Y$ between topological spaces $X,Y$ is defined to be
continuous if $f^{-1}(V)$ is open in $X$ for all open $V \subset Y$,
open if $f(U)$ is open in $Y$ for all open $U \subset X$...
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Characterization of Continuous, Closed and Open maps
I've been strugrilling trying to prove the following results from Lee's book on Topological Manifolds.
Proposition 1. Let $f:X\to Y$ be a function between topological spaces.
$f$ is continuous if ...
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An application of open mapping theorem in linear algebra.
Open mapping theorem,which is taught in functional analysis,states that if $X$ and $Y$ are two Banach spaces and $T:X\to Y$ be a surjective bounded linear operator ,then the map $T$ is open.Now I ...
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Is the expectation operator an open map?
Let $\Delta [0,1]$ denote the set of Borel probability measures on $[0,1]$. $\Delta [0,1]$ endowed with the Prokhorov metric is a metric space, as we know. My question is, does the expectation ...
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Is there a continuous open surjective map from the 2-sphere to the 2-torus?
Is there a continuous open surjective map from the 2-sphere $S^2$ to the 2-torus $S^1 \times S^1$?
[Some thoughts: Since both spaces are compact, any continuous surjective map is a quotient map. ...
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A question related to the closedness of a map between two topological spaces
Consider $\mathbb{R}$ with the Euclidean topology. Suppose we have an equivalence relation on $\mathbb{R}$ such that the equivalence classes are $\mathbb{Z}$ and single non-integer points. Let $q: \...
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How to characterize open sets in the unit circle
At the end of the second chapter of Munkres' "Topology" there are some supplementary exercises concerning topological groups. One such group is $(S^1, \cdot)$. My problem is that the unit ...
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Example of a function such that the property $ \forall U \in \tau, f^{-1}(f(U)) \in \tau$ doesn't hold.
What continuous function $f$, set and topology $\tau$ could serve as a counter-example of the following property : $ \forall U \in \tau, f^{-1}(f(U)) \in \tau$ ?
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Bounded surjective linear map from $L^p$ to $L^r$ with $r < p$
I am thinking about the following problem. Let $1 \leq r < p \leq \infty$, and consider the spaces $L^r([0,1])$ and $L^p([0,1])$. A routine application of Holder shows that $L^p([0,1]) \subset L^r([...
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Given an open and inyective function $f:(X,T)\rightarrow (Y,S)$, and given that $(Y,S)$ is separable, find if $(X,T)$ is separable.
Hi I was wondering if I got this right, this is my try:
Let $A$ be an open set of $T$, then because $f$ is open $f(A)$ must be an open set of $S$.
Now, given that $(Y,S)$ is a separable space there ...
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Determine if analytic mappings from $\mathbb{C}\setminus \{0\}$ and $\mathbb{C}\setminus [0,\infty)$ to open unit disk exist and if so, find them...
This is a question from a previous complex analysis qualifying exam that I'm working through to study for my own upcoming exam. I'm not sure how to approach it. I've included my thoughts on it, but I'...
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$\forall \alpha\in\mathscr{A},\; X_\alpha\cong Y_{\varphi(\alpha)}\implies \prod_\alpha X_\alpha\cong \prod_\beta Y_\beta$ (Homeomorphisms).
The Unclear Claim
I'm having trouble understanding the following statement from Topology by James Dugundji chapter IV:
$\mathbf{2.6}$ Corollary. Let $\{X_\alpha\mid \alpha\in\mathscr{A}\}$ and $\{Y_\...
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A homeomorphism essentially takes a topological space into open subspace of the real? [closed]
I've been question myself if the subspace of the real where we map a topological space needs to be necessarily open since the function that do this map is bijective and continuous, someone can help me ...
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The property of the saturated sets
I am trying to solve Exercise 3.59 from Introduction to Topological Manifolds, by John Lee.
Let $q: X \rightarrow Y$ be any map and $U \subseteq X$.
Given "If $x\in U$, then every point $x' \in X$...
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Question about continuous but neither open nor closed function on topological space X
Define a function from topological space $\mathrm{X}$ into topological space $\mathrm{X}$, $f:\mathrm{X}\rightarrow \mathrm{X}$ such that $f$ is continuous but neither open nor closed
My attempt
I ...
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Are all identification maps on a topology open? (proof verification)
I came up with the following proof yet feel like it's wrong (justification for this feeling at the end):
Thorem. Let $(X,\mathscr{T}_X), (Y,\mathscr{T}(p))$ be spaces and $p\colon X\to Y$ an ...
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Continuous functions carry compact sets into compact sets. Proof Attempt
I'm reading loomis and this is left as an exercise but I'm not convinced I'm right. Can you please verify my proof and help me finish it? Also, what is the difference between compact and sequentially ...
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Question in Folland's Proof of Open Mapping Theorem
The Open Mapping Theorem states:
Let $X$ and $Y$ be Banach spaces. If $T \in L(X, Y)$ is surjective, then $T$ is open.
Below is a proof from Folland's text. I am able to follow up his proof until a ...
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Show that the range $R(T)$ is not closed in $l^2$ norm for $T:(x_1, . . . , x_n, . . .) → (x_1, . . . ,\frac{1}{n}x_n, . . .)$
Here is the question and the answer:
I would like to ask in the first and second line, how does $R(T)$ being closed in $l^2$ and the open mapping theorem imply the inverse mapping $S$ being bounded.
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Show that $\dim(Y)<\infty$, application of the Hahn-Banach theorem.
$\textbf{The question is}$
Let $X,Y$ be a Banach space, $T:X\rightarrow Y$ an surjective linear transformation.
If $\exists D\subseteq Y$ compact: $T(B(0,1))\subseteq D$ then $\dim(Y)<\infty$
$\...
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Find $X,Y\subseteq \Bbb{R}$ such that there are $f\colon X\to Y$ and $g\colon Y\to X$, both bijective and continuous but $X\not\cong Y$.
I came across the following problem:
Let $\Bbb{R}$ be the euclidean space. Find $X,Y\subseteq \Bbb{R}$ such that there are maps $f\colon X\to Y$ and $g\colon Y\to X$, both bijective and continuous ...
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Application of the open function theorem (BANACH) with $f\not\equiv 0$.
THE PROBLEM STATEMENT IS AS FOLLOWS :
Let $X$ be a Banach space over $\mathbb{R}$ and $f:X\rightarrow \mathbb{R}$ a continuous linear functional and $f\not\equiv 0$ then $f$ is open.
DEMONSTRATION:
it ...
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Can an injective contraction take a set with nonempty interior to one with empty interior?
Suppose that $(X,d)$ is a complete metric space and that $f \colon X \to X$ is an injective contraction. That is, there exists $K \in (0,1)$ such that $0 < d(f(x),f(y)) \le K d(x,y)$ for all $x,y \...
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