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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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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 ...
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2answers
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continuous image of a locally compact space is locally compact

Is continuous image of a locally compact space is locally compact? Let $X$ be locally compact(l.c.).Let $f:X\to Y$ is continuous and surjective. A space $X$ is locally compact if for each $x\in X$ ...
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1answer
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When quotient map is 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?
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2answers
837 views

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 ...
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2answers
63 views

If $f :{\Bbb R}^2 \to {\Bbb R}^2$ is a function satisfying $\| f(x) - f(y) \| \geq 3\|x-y\|$ , then $f$ is an open mapping and generalizing it

Let $f : {\Bbb R}^2 \to {\Bbb R}^2$ be a function such that $\| f(x) - f(y) \| \geq 3\|x-y\|$ ($\|\cdot\|$ denotes the usual metric in ${\Bbb R}^2$). Then I was trying to show that $f$ is an open ...
10
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1answer
331 views

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 ...
5
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1answer
269 views

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 $A$ ...
5
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1answer
1k views

Open and Closed mapping Examples

I am looking for three mappings f:X to Y any set of topology on X or Y. so very flexible. Can you help me find an example of a function that is (a) continuous but not an open or closed mapping (b) ...
4
votes
2answers
204 views

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 ...
13
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1answer
397 views

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 ...
4
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1answer
138 views

Is exponentiation open?

Already for $2\times 2$ matrices the exponential map is not open. However, the diagonalization trick does not work for algebras of functions. Hence the question Is the map $f\mapsto \exp(f)$ open on ...
2
votes
1answer
168 views

Local homeomorphism iff discrete fibers and open and [?]

A continuous map does not tear. A local homeomorphism also "locally does not glue". This is captured in part by the fact it has discrete fibers: the points which are glued are isolated from each other ...
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vote
1answer
441 views

Show that a continuous, onto function from a Compact Space is an open map.

$X$ is compact and $f$ :$X$ $ \to $ $Y$ is continuous and onto. Then show that $f$ is an open map. i.e. Every open set of $X$ is mapped to an open set of $Y$.