# 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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### 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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### 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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### 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 ...
1 vote
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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 ...
1 vote
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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. ...
1 vote
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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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### 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 ...
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 \...