# 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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### 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 $g:R^2 →R$ such that $g(x,y)=\sin y+y+e^x −1$ $\forall (x,y)∈R^2$ 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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### “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 ...