# 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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### 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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### Find continuous functions $f,g$ such that $g\circ f$ is closed and continuous but neither $g$ nor $f$ is closed map.

Find continuous functions $f,g$ such that $g\circ f$ is closed and continuous but neither $g$ nor $f$ is closed map. Find continuous functions $f,g$ such that $g\circ f$ is open and continuous ...
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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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### Compact open operator between Banach spaces

Let $X,Y$ be Banach space, $Y$ infinite dimensional. Show that no $T \in \mathcal{K}(X,Y)$ is open. By definition $T$ is open if and only if $\exists r >0$ such that $B_Y(0,r) \subset T(B_X(0,1))$ ...
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### Let $Y$ be a finite dimensional normed space, $X$ a normed space, and $T: X \to Y$ a surjective linear operator. Show that $T$ is an open mapping.

Let $Y$ be a finite dimensional normed space, $X$ a normed space, and $T: X \to Y$ a surjective linear operator. Show that $T$ is an open mapping. I think if I can show that $T(B_X)$ contains an ...
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### Discontinuous surjective linear map which is not open

The following statement is true: Assume that $X$ and $Y$ are topological vector spaces where $Y$ is finite-dimensional Hausdorff, if $A:X\rightarrow Y$ is a continuous surjective linear map then $A$ ...
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### Product maps are open maps

I am trying understanding product maps better. Suppose that we have open maps$$f_1:X_1\rightarrow Y_1, f_2:X_2\rightarrow Y_2$$ Then, if $U_i\subseteq X_i \Rightarrow f_i(U_i)\subseteq Y_i$ for $i=1,2$...
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### Explicit/constructive example of open maps that are not continuous (especially from R to R)?

TLDR: I'm looking for an explicit map that is an open map but not continuous. The context my question arose was when learning the topological definition of continuous function. I made some progress ...
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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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### Why does this statement hold?

I have seen the following statement: Let $G\subset \mathbb{R}^n$ be open, bounded and $f:\overline{G}\rightarrow \mathbb{R}^n$ a continuous and open map. Then $\|f\|$ gets its maximum on the ...
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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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### Map $F$ open and $F(\mathbb{R})$ is a submanifold

Let $F: \mathbb{R} \rightarrow S^1 \times S^1$ defined by $$F(t) = ((\cos2\pi t, \sin2 \pi t), (\cos2\pi \alpha t, \sin2 \pi \alpha t))$$. For which values ​​of $\alpha$ the map $F$ is open? For ...
Let $X$ be a topological space and $Y$ a subset of $X$. Write $i: Y \to X$ for the inclusion map. Choose the correct statement: If $i$ is continuous, then $Y$ has the subspace topology. If $Y$ is an ...
I recently read that, given topological spaces $S,T$ and a map $f:S\rightarrow T$, for $f$ to be open it is sufficient to show that for a certain subbasis $C$ of $T$ and all (open) sets $A\in C$ holds ...