Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.
share|cite|improve this question
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
It might help to have a more specific title. – Hui Yu Apr 2 '13 at 14:44
  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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.