# Questions tagged [closed-map]

In topology, a closed map is a function between two topological spaces which maps closed sets to closed sets. That is, a function f : X → Y is closed if for any closed set U in X, the image f(U) is closed 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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### How can this open map characterization be explained or interpreted?

I hope you're having a good day. I'm an undergrad mathematics student, I took a general topology course a few months ago, and I'm now reviewing topological spaces to prepare for functional analysis. I ...
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### Surjective, closed local homeomorphism with finite fibers is a covering map?

Let $E,X$ be connected, locally path connected topological spaces and $\pi: E \to X$ a surjective, closed local homeomorphism with finite fibers. Is $\pi$ necessarily a covering map? I know that this ...
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### How to show directly that the function $f \colon \mathbb{R} \longrightarrow \mathbb{R}$, $x \mapsto x^2$ is a closed map?

Let the function $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ be defined by $f(x) := x^2$ for all $x \in \mathbb{R}$, where the set $\mathbb{R}$ of real numbers has the usual topology. Then $f$ is ...
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### Let $(X, \mathcal{T})$, $(Y, \mathcal{U})$ be topological spaces, and let $f: X \rightarrow Y$ be a bijection. Then $f$ is open iff $f$ is closed

Here's my proof of the above statement. Since $f$ is open, it maps open sets in $\mathcal{T}$ to open sets in $\mathcal{U}$. Let $U \in \mathcal{T}$ be a closed set. So, $X \setminus U$ is an open set....
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### A ball in $H^m(\Omega)$ is closed under $L^2$-convergence [closed]

I read it in a paper in which the author aims to claim that if a sequence $\{u_n\}$ is bounded in $H^m(\Omega)$ (here $\Omega$ is a bounded domain), and $u_n$ has a limit $u$ in $L^2$ norm, then $u$ ...
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### Example of a closed map which is not continuous.

A map $F: X \to Y$ is said to be closed if $x_n \to x$ in $X$ and $F(x_n) \to y$ in $Y$ implies $y = F(x)$. As confusing as the definition is, I cannot seem to understand how a closed map is not ...
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### For a closed subset $F ⊂ X × Y$ , the image $π(F )$ need not be closed in Y

I have the following question from an exam If $(X, d_X)$ is compact, show that every sequence in $X$ has a subsequence converging to a point of $X$. Deduce that the projection map $\pi$ then has the ...
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### A linear transformation is open map if and only if surjective and closed map if and only if injective

Q.If $d$ and $e$ are positive integer and $T:R^{d} → R^{e}$ be a linear transformation then (a) $T$ is open map if and only if $T$ is surjective (b) $T$ is closed map if and only if $T$ is either zero ...
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### Inverse image of a point under a continuous surjective closed map :

Let $f:X\rightarrow Y$ be a continuous surjective closed map and $X$ is a normal space. Let there exist an open set $U\subset X$ such that $f^{-1}\{ y \}\subset U$ then show that there exist an open ...
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### Mapping a hemisphere onto the unit circle

How can I approach attempting to map a hemisphere onto the unit circle such that the meridians become arcs of a circle through $(0,1)$ and $(0,-1)$ and are evenly spaced around the x-axis (which is ...
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### How can I show that $f\mapsto f'$ is a closed map? [duplicate]

Let $X=C^1([0,1])$ and $Y=C([0,1])$ both equipped with $\|f\|=\max |f|$. Then consider $$D:X\rightarrow Y:~~f\mapsto f'$$ I want to check that $D$ is a closed map. By definition we need to show that ...
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### $S_1$: compact, $S_2$: any topological space. Then $S_1\times S_2\ni (x,y)\mapsto y\in S_2$ is a closed mapping.

I am reading "Introduction to Set Theory and Topology" (in Japanese) by Kazuo Matsuzaka. Definition: Let $(S,\mathcal{O})$ be a topological space. Let $x\in S$. Let $V\subset S$. If $x$ is ...
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### Could you suggest a textbook for learning basic properties of spectrum of closed operators?

I wish to learn the very basic properties of the spectrum of Closed operators. I may use these properties in the research in the Fluid dynamics/Differential equations. My attempts:- I searched in ...
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### Example of closed map from $T_2$-space onto non-$T_2$ quotient? [duplicate]

What is an example of a continuous closed surjection $f : X \rightarrow Y$ from a $T_2$-space $X$ to a space $Y$ that is not $T_2$?
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### $f: X \rightarrow Y$ injective, closed. Prove that for $U \in \mathcal{T}_x$ and $M$ ⊆ $Y$, ∃ $V \in \mathcal{T}_y$ s.t $M$⊆$V$ and $f^{-1}(V)$ ⊆ $U$

Let $(X,\mathcal{T}_x), (Y,\mathcal{T}_y)$ be topological spaces and $f: X \rightarrow Y$ be an injective and closed map (i.e. it maps closed sets to closed sets). Let $U$ be an open subset of $X$ and ...
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### Question about continuous but neither open nor closed function on topological space X

Define a function from topological space $\mathrm{X}$ into topological space $\mathrm{X}$, $f:\mathrm{X}\rightarrow \mathrm{X}$ such that $f$ is continuous but neither open nor closed My attempt I ...
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### The graph of a linear operator is closed but not continuous (closed graph theorem)

$\textbf{The question is}$ Let $X:=C^{1}[0,1]$, $Y:=C^{0}[0,1]$ both equipped with the $\left \| f \right \|:=\begin{matrix} Sup & \left | f(x) \right |\\ x\in [0,1]& \end{matrix}$ and ...
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### Given domain $[0,T]$, can we have a 2nd order ODE describing $[0,T/2]$...?

If we have a whole time domain from $[0,T]$ can we have a second order ODE describing $[0,T/2]$ and then a first order ODE describing $[T/2,0]$ ensuring that the solutions of the ODE match up at $0$ ...
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### An example of a closed quotient map that is not open

We consider the map $$f\colon X:=[0,2\pi]\times [0,1]\to Y:=S^1\times [0,1],\quad\text{defined as}\quad f(t,s)=((\cos t, \sin t), s).$$ $f$ is a closed quotient map and corresponds to the equivalence ...
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### How to prove T is a closable operator

Let $T:H\to H$ a densely defined operator, with $H$ a Hilbert space such that: $$Re(x,Tx)\geq 0, \forall x\in Dom(T)$$ I want to prove that $T$ is a closable operator, that means... that there exists ...
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### Confusions about proof regarding proper maps

I am currently trying to prove that if $f:X\rightarrow Y$ is a closed continuous map between topological spaces such that $f^{-1}(y)$ is compact for all $y\in Y$ then f is proper. In this sense $f$ ...
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