# Questions tagged [retraction]

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### Retract of noncompact surface to its boundary?

Suppose $M$ is a connected, noncompact 2-manifold, and its boundary $\partial M$ is a circle. What's the simplest way to show there is a retraction $r: M\rightarrow \partial M$? Here are some ...
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### Deformation Retraction and Projection/Closest Vector

How do I compute the projection/closest vector to a subset? I have been thinking about this for far too long without any progress. If it helps, I am working in $\Bbb{R}^2$, but I would like formalue ...
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### Questions about the proof sketch of “$\{x\}$ is a deformation retract of $\overline{St}(x)$”

Munkres Topology Section 83 First question: Is "obvious deformation" the straight line homotopy $F(b,t) = \overline{St}(x) \times I \to \overline{St}(x)$? Second question: Is the "result" in "This ...
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### strong deformation retracts & neighbourhood deformation retracts

Let $X$ be a topological space and $A \subset X$ be a closed subspace. Assume that $A$ is a strong deformation retract of $X$. Is it true that for every open subspace $U \subset X$, $A \subset U$, ...
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### Retractions and local finiteness: Are these circles a retract of $\mathbb R^2$?

Let $N$=$\{1+\frac 1 n\;|\;n\in\mathbb N\}\cup\{1\}$. I'm interested in whether the space $$A=\cup_{n\in N}A_n \text{ where } A_n:=\{(x,y)\in\mathbb R^2\;|\;(x-n)^2 + y^2 = n^2, x\leq 1\}$$ is a ...
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### A compact convex subset is a strong deformation retract

Show that every compact convex subset of $\mathbb{R}^n$ is strong deformation retract. I don't really know how to approach to this. Any help would be very appreciated. Thanks!
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### Say if the following function is nulhomotopic

$f:\mathbb{R}^2 -\{0\} \to \mathbb{R}^2 -\{0\}$ such that $f(x,y)=(x^2, y)$ I believe this isn't nulhomotopic since I can't pass through 0, but I don't know how to justify this. I think I should ...
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### Prove: If $n \ge 0$, then $S^n$ is not a retract of $D^{n+1}$ using homology functors.

Prove: If $n \ge 0$, then $S^n$ is not a retract of $D^{n+1}$ using homology functors. The book I'm reading says: For each $n \ge 0$, there is a homology functor $H_n$ with the following properties:...
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### Is the intersection of a descending chain of group retracts a retract?

Let $G$ be a group. A subgroup $H\subset G$ is said to be a retract if there exists surjective homomorphism $\pi:G\to H$ s.t. $\pi\circ\iota=Id_{H}$ with $\iota$ being the inclusion of $H$ in $G$. So ...
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### Retract of $D^n \times [0,1] \times [0,1]$

Question: Show that $D^n \times [0,1] \times \{0\} \cup S^{n-1} \times [0,1] \times [0,1]$ is a retract of $D^n \times [0,1] \times [0,1]$. Attempt: I have proved a more intuitive retraction, ...
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### Prove that two quotients of a bounded cylinder are not homeomorphic

Let $C$ be the cylinder $\{(x,y,z)\mid x^2+y^2\leq 1, |z|\leq 1\}$. Consider two relations $R$ and $R'$ on $C$ which identify a point such that $x^2+y^2=1$ with $(-x,-y,z)$ and $(-x,-y,-z)$ ...
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### What is special about retraction mapping?

What is special about the retraction mapping? Can't we always find such a mapping, namely identity map of $X$. Then every space $A$ will be a retract of $X$. EDIT: Do we need retraction to be ...