# 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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### An isometric mapping $f:(X,d) \to (X, d)$ is an open mapping

In general, the result doesn't hold. For example $f: \mathbb{R} \to \mathbb{R}^2$ with $f(x)=(x,0)$ [Euclidian Norm]. Now my attempt to prove the restricted case goes as follows: Let $V \subset X$ ...
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### Rudin functional analysis, theorem 2.11 (Open mapping theorem)

Going through the proof of such theorem, there are few bits I don't understand. I'll write down the proof and in in the middle I'll add some comments. Statement: Suppose (a) X is an F-space, ...
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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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### Analytic function being open

I was reading this question where it says that an analytic complex function $f:D \to \mathbb{C}$ are open mappings when $f′(z)$ is never zero. I´m not clear on two things, pardon me if they are ...
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### Why is the inverse of a bounded bijective operator continuous?

I'm facing some troubles with the following theorem, Let $X,Y$ be Banach spaces and let $T \in B(X,Y)$. I want to show that if $T$ is bijective then its inverse is continuous. Now if $T$ is bijective ...
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### Closed Range Convolution Operator

Does there exists a nontrivial $f \in L^1(\mathbb{R})\cap L^\infty(\mathbb{R})$ such that $f\ast L^\infty(\mathbb{R})$ is a closed subspace of $L^\infty(\mathbb{R})$? I couldn't find any good ...
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### Submmersions: $|f(x)|$ do not assume maximal value

I am studying analysis and I have had a lot of uncertainties. For instance, I cannot solve this exercise: If $f:U\rightarrow\mathbb{R}^3$ has class $C^1$ and rank $3$ in all of the points of the open ...
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### Image of locally compact space under a continuous open map

In Wayne Patty's book titled Foundations of Topology, local compactness is defined as follows: $X$ is locally compact at a point $p$ in $X$ provided that there is an open set $U$ and a compact set $K$ ...
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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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### Queris related to “An arbitrary product $X=\prod_{\alpha \in J} X_\alpha$ is connected iff $X_{\alpha }$ is connected for each $\alpha\in J$.”

The following proof of An arbitrary product $X=\prod_{\alpha \in J} X_\alpha$ is connected iff $X_{\alpha }$ is connected for each $\alpha\in J$ is provided by my professor.I've some ...
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### $f$ is open iff $f(U)$ is open for every $U$ open set of a fixed point $x \in X$ in a homogeneous space

Let $X$ be a homogeneous space (i.e. for every $x,y \in X$ there is a homeomorphism $\phi : X \to X$ such that $\phi(x)=y$) and $x \in X$ a fixed point, if $f$ is a function such that $f(U)$ is open ...
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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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### Quotient map from topological group onto left cosets is open

I've been trying to prove the following exercise from "James Munkers'-a first course in topology": Let $G$ be a topological group, and let $H\leq G$. Induce the left cosets, $G/H$, with the quotient ...
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### How can I verify that this standard map is symplectic?

Show that the following standard map is symplectic $$I(t+1) = I(t) + K\sin\theta(t)$$ $$\theta(t+1) = \theta(t) + I(t+1) \bmod 2 \pi$$
### Showing $\forall q\in\mathcal{P}_Y$ there exists $p\in\mathcal{P}_X:q\circ A\le p\implies A:X\to Y$ continuous
Let $(X,\mathcal{P}_X)$ and $(Y,\mathcal{P}_Y)$ be locally convex topological vector spaces with topologies induced by the families of continuous seminorms $\mathcal{P}_X$ and $\mathcal{P}_Y$ ...
$\newcommand{\Spec}{\mathrm{Spec}}$ Let $R \rightarrow S$ be a faithfully flat (unital) homomorphism of commutative rings. Does it follow that the corresponding map of topological spaces \$\Spec S \...