# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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 ...