Mathematics Stack Exchange
Mathematics Stack Exchange

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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Projection is an open map

Let $X$ and $Y$ be (any) topological spaces. Show that the projection $\pi_1$ : $X\times Y\to X$ is an open map.
Alisha
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Open maps which are not continuous

What is an example of an open map $(0,1) \to \mathbb{R}$ which is not continuous? Is it even possible for one to exist? What about in higher dimensions? The simplest example I've been able to think of is the map $e^{1/z}$ from $\mathbb{C}$ to…
math student
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When is a quotient map open?

Quotient map from $X$ to $Y$ is continuous and surjective with a property : $f^{-1}(U)$ is open in $X$ iff $U$ is open in $Y$. But when it is open map? What condition need?
Oh hyung seok
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Example of quotient mapping that is not open

I have the following definition: Let ($X$,$\mathcal{T}$) and ($X'$, $\mathcal{T'}$) be topological spaces. A surjection $q: X \longrightarrow X'$ is a quotient mapping if $$U'\in \mathcal{T'} \Longleftrightarrow q^{-1}\left( U'\right) \in…
Luc M
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Intuition for an open mapping

What is an intuitive picture of an open mapping? The definition of an open mapping (a function which maps open sets to open sets) is simple sounding, but it's really not as easy to picture as the simple language would suggest. When I think of…
echinodermata
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Show that an open linear map between normed spaces is surjective.

Let $X,Y$ be normed spaces and $T:X\to Y$ is an open linear map. Show that $T$ is surjective. In order to show $T$ is surjective let's take $y_0\in Y$ and assume the contrary that $Tx\neq y_0\forall x\in X$. Now taking $x_0\in X\implies Tx_0\neq…
Learnmore
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A bijective continuous map is a homeomorphism iff it is open, or equivalently, iff it is closed.

Wikipedia states that "a bijective continuous map is a homeomorphism if and only if it is open, or equivalently, if and only if it is closed.". How do we prove this fact? I can prove the obvious direction, but im unsure how to proceed the other ways
terry
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Extending open maps to Stone-Čech compactifications

Let $X$ be a Čech-complete space, and $Y$ a paracompact space. Suppose $f\colon X\to Y$ is a continuous and open surjection. Since $Y$ is completely regular we have that $\beta(Y)$ is homeomorphic to $Y$ as a dense subset of $\beta Y$ (the…
Asaf Karagila
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Find $X,Y\subseteq \Bbb{R}$ such that there are $f\colon X\to Y$ and $g\colon Y\to X$, both bijective and continuous but $X\not\cong Y$.

I came across the following problem: Let $\Bbb{R}$ be the euclidean space. Find $X,Y\subseteq \Bbb{R}$ such that there are maps $f\colon X\to Y$ and $g\colon Y\to X$, both bijective and continuous but $X\not\cong Y$ ($X$ not homeomorphic to…
Sebastián P. Pincheira
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When is the quotient space of a second countable space second countable?

I am a bit confused about this concept because I have read that the quotient space is second countable if the quotient map is open. However, I thought the definition of a quotient map was a surjective, continuous, open mapping. Suppose that $X$ is…
user190570
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a covering map is open?

$E,B$ are topological spaces and let's say that $p:E\to B$ is a covering map. Is $p$ open? i tried to show it as follows: let $U$ be an open set in $E$, and now for every $x\in U$, $p(x)\in B$. $p$ is a covering map so there is a open neighborhood…
sha
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Attempted proof of an open mapping theorem for Lie groups

The Classical open mapping theorem for Banach spaces tells that if $T:X \to Y$ is a continuous surjective linear map, then it is open. I have attempted to essentially "adapt" the proof for Lie groups: Let $G,H$ be connected Lie groups (embedded in…
Andres Mejia
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Is a certain restriction of an open map open?

Let $p:X\rightarrow Y$ be an open map and let $A$ be a subspace of $X$. Then, is it true that $p|_A:A\rightarrow p(A)$ is open? I think so, but am struggling to show it. My thoughts: Let $O$ be an open set in $A$. Then, since $A$ has the subspace…
user322548
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Characterizing continuous, open and closed maps via interior and closure operators

A function $f :X \to Y$ between topological spaces $X,Y$ is defined to be continuous if $f^{-1}(V)$ is open in $X$ for all open $V \subset Y$, open if $f(U)$ is open in $Y$ for all open $U \subset X$, closed if $f(C)$ is closed in $Y$ for all…
Paul Frost
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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 where $ f $ is injective we get a…
Nemesis
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