determine which of the following spaces are contractible Determine which of the following spaces are contractible?
(a) Unit interval $I=[0,1]$
(b) $\mathbb{S^2}$$\setminus$ {$p$},where $\mathbb{S^2}$ is a $2$-sphere and $p$ is any point on $\mathbb{S^2}$
(c) Any solid or hollow cone in $\mathbb{R}^3$.
(d) The subspace {$0$} $\cup$ {$1/n:n \in \mathbb{N}$} of real line.

Intuitively I can say that:-
(a) it is contractible.
(b) No idea.
(c) the solid cone is contractible but the hollow one is not.
(d) No idea.
Can someone help me, please? Please provide me with the proof of (a) & (c) also, if I am right. Thanks for your help.
 A: (a) It is. Notice that the point $\{0\}$ is a deformation retract of $[0,1]$.
(b) It is. $S^2\backslash\{p\}$ is homeomorphic to a disk.
(c) Solid is, hollow not (not that the hollow one has the circle $S^1$ for a deformation retract).
(d) It is not, since it is not connected.
Now try to make those formal.
A: Just adding a little bit... (a) is contractible, since you can link the identity map with a constant map by means of the homotopy $f_t(x) = tx$. (b) is also contractible (however, the whole sphere is not) For (c), as it was already mentioned, solid cone is contractible, but when you consider this hollow cone, it depends whether you consider it with or without bottom (in $\mathbb{R^3}$) without bottom it is contractible, with - no. Well, a kind of an intuitive way to determine whether a space is contractible is to think of how it could be deformed continuously into a single point (but only along its own points) without tearing it apart (for instance, you cannot do it with a circle on a plane, but as soon as you take any single point away from this circle (tear it apart), you can collapse the resulting space into a point)...
