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Can we help me next tags:

  1. Let $Y$ vector topological space and $A \subseteq Y$ convex set. Prove that any two continuous mappings $f,g : X \to A$ homotopic, where $X$ is an arbitrary topological space.
  2. Prove that every interval $(a, b)$ homotopy equivalent point.
  3. Let $X$ be a contractible space. Prove that the space $X$ every two times with the same beginning and end homotopic (rel $\{0,1\}$).
  4. Prove that two topological spaces one of which is attached, and the other can not be unlinked homotopy equivalent.
  5. Show that the fundamental group of the space $\mathbb R^n$ , $n\geq 1$, trivial.
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Can you use more enter? I don't understand 3. Please also provide, how far could you get with the solutions. Use $ for mathematical formulas, \subseteq for subset and \to for arrow, and \{ and \} for the set parenthesis. –  Berci Apr 2 '13 at 10:47
    
Edited. I'm only guessing at the symbol between A and Y, since it came across as a blank square. –  Thomas Andrews Apr 2 '13 at 10:52
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It might help to have a more specific title. –  Hui Yu Apr 2 '13 at 14:44
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1 Answer

  1. Let $f,g: X \to A \subseteq Y$ be continuous. Then $F(x,t) = tf(x) + (1-t)g(x)$ where $t \in [0,1]$ defines a homotopy from $g$ to $f$.

  2. Let $(a,b) \subseteq \mathbb R$. Then $(a,b)$ is a convex set. Let $f(x) = \mathrm{id}: (a,b) \to (a,b)$ be the identity map and let $g(x) = r$ for some $r \in (a,b)$. Now apply 1. to obtain the desired result.

  3. Let $X$ be homotopy equivalent to a point and let $p,q: [0,1]\to X$ be two paths in $X$ with $q(0) = p(0) = q(1) = p(1) = x_0$. Let $H$ be the homotopy from $\mathrm{id}_X$ to $f(x) = x_0$. Then define a homotopy $h$ from $p$ to $q$ relative $\{0,1\}$ as $h(x,t) = H(x,2t) \circ f$ for $t \in [0,1/2]$ and $h(x,t) = H^{-1}(x,2t - 1) \circ g = H(x,2- 2t)\circ g$ for $t \in [1/2,1]$.

  4. Can you define what attached means please?

  5. The fundamental group of a contractible space is trivial. Show that $\mathbb R^n$ is contractible.

Hope this helps.

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