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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A continuous open surjective map $f$ with compact fibers preserves Hausdorffness.

Let $f:X\to Y$ be an open, surjective, continuous map between two topological spaces $X$ and $Y$ with compact fibers. Let $X$ be a Hausdorff topological space. Then $Y$ is Hausdorff. I wanted to show ...
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On Banach-Schauder theorem (open-mapping)

Theorem Let $X$, $Y$ be Banach spaces and $T \in {\cal B}(X,Y)$ a surjective linear map. Then $T$ is an open map (that is, it sends open sets in open sets and the inverse $T^{-1}$ is continuous). Now,...
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An application of Open Mapping Theorem on the identity operator [duplicate]

I have an assignment in the Functional Analysis Course to submit which the reference book is Conway's Functional Analysis. I couldn't reach a solution nor did I find on the Internet. But I found that ...
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Given a product topology X = $\prod_{i \in I}X_i$ , check: if E is closed in X, then $\pi_i(E)$ is closed in $X_i$

Given: A family of topological spaces: $\{(X_i, \mathcal{T_i})\mid i \in I\}$ $X = \prod_{i \in I}X_i$ is a product topology with topology defined as $\mathcal{T}$ $\pi_i: X \rightarrow X_i$ is ...
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Behaviour of ${f}^{-1}$ under continuous map

Let $f : \Bbb R \to \Bbb R$ be a continuous function. Which of the following is/are always true ? ${f}^{-1}(A)$ is open for all open sets $A \subseteq \Bbb R$ ${f}^{-1}(A)$ is closed for all closed ...
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Proving operator image is closed

Let $T:X\to Y$ be a linear bounded and surjective operator between Banach spaces $X$ and $Y$. I want to prove that if $A+\ker T$ is closed then $T(A)$ is closed. I tried using the open mapping theorem ...
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Proving this map to be open and that the quotient space is not Hausdorff

This question is from Section 5.1 ($T_0 - T_2$ spaces) of Wayne Patty's topology and I am struck on it. So, I am asking for help here. Let $I =[0,1]$ and define $x \sim y$ provided that $x-y$ is ...
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Function that is closed but not open and not continuous

So, I am trying to find a function with requirements of being closed but not open and not continuous which is defined on the real numbers and the usual topology. I was able to find many piecewise ...
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How can I show that $T(O)$ is not open in $Y$? where we have that $O = [{x \in X : ||x||_X < 1}]$. $X:[ {x = (x_1, x_2, x_3, . . .) : x_n ∈ \mathbb{K}, \sum_{n=1}^{\infty}|x_n|< \infty}]$ and $||... 1answer 52 views A surjective linear mapping$L$from$X$to$Y$is open iff there is a constant$c$for each$y=Lx$satisfying$\|x\| \leq c \|y\|$So I have to show both directions. For the if direction, I began as below. Suppose such a$c$exists for each pair of$x \in X$and its corresponding$y = Lx \in Y$. I have to show that the image of ... 1answer 44 views $f$Open, Closed and continuous but not a local homeomorphism [closed] Give an example of topological spaces$X$and$Y$and a map$f: X \to Y$that$f$is an open, closed, and continuous but not a local homeomorphism. (The map and the topological spaces have to satisfy ... 1answer 73 views Prove that if$f$is a continuous map which maps every open set to an open set then$f$is monotonic. My attempt:If the function is not monotonous then we can consider points$a < b$and$b <c$such that$f(a)<f(b)$and$f(b)>f(c)$.My intuition behind this was to consider a subset$[a,c]$... 1answer 24 views How to prove the continuity of the following Homeomorphism? Let$f$be a function from the one-point compactification of$\mathbb{R}$to the circle defined as follows$t\mapsto (x,y) = \begin{cases} \phantom{\lim\limits_{t\to\infty}} \left( \dfrac{1-t^2}{1+t^2}...
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let $f:(0,1] \rightarrow \mathbb{R^2} , f(t)=\left(t \cos \left( \frac{1}{t}\right), t \sin \left(\frac{1}{t}\right)\right )$ is $f$ an immersion, embedding or proper embedding? $f$ is an immersion ...
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open map between smooth manifolds

Let $N, M$ be two smooth manifolds and $f: N \rightarrow M$ a smooth immersion. (i)-If $f$ is a homeomorphism on its image the $f(N)$. Then $f(N)$ is a submanifold. (ii)- If the dimension of $N, M$ ...
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