Is every contractible space a cone?

A fun counterexample: written in 1-dimensional strokes, the letter $\pi$.

A whole family of counter examples comes from finite spaces. Clearly, no finite nonempty space can be the cone of anything. But contractible finite spaces exist (there is even an entire relevant book, "Algebraic topology of finite topological spaces and applications", Barmak).

One way to quickly convince yourself that the answer must be 'no' is that cones are strongly related to $[0,1]$ (by construction!). So, any cone will have to admit some $\mathbb R$-like properties. However, contractibility is a very refined topological property and it is highly unlikely that just because a space is contractible that it will be strongly related to the real numbers. Maybe this counts as a non-constructive way to see that counterexamples "should" exist.

For an explicit non-finite example, consider http://en.wikipedia.org/wiki/House_with_two_rooms

Why are the order-of-operations conventions good?

Convolution intuition: clarifying Terence Tao's "blurring"/"fuzz" interpretation

Uniform convergence and weak convergence

A conjecture: for all $n\in\mathbb{N}$, the least $k>1$ such that $\phi(k)\geqslant n$ is a prime

Show that the sequence ${a_n}$ converges where $a_n = \sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt{n}}}}$ for $n\geq 1$.

What is a concrete example of why one wants to have a *derived category* in algebraic geometry?

Chinese Remainder Theorem clarification

Making a convex polyhedron with two sheets of paper

The function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$.

Show $\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$ diverges

Bayes, two tests in a row