TITLE: Difference between homotopy equivalence and based homotopy equivalence?
QUESTION [9 upvotes]: As the question suggests, what is the difference between homotopy equivalence and based homotopy equivalence?

REPLY [3 votes]: A standard example is to take $H$ to be the Hawaian Earring, which has a wild point at $0$. The cone $CH$ on $H$ is contractible, as is any cone;  but to contract it you have to move the base "up" the cone to the vertex; if the base point is taken as the wild point at the bottom of the cone, then you cannot contract $CH$ keeping that point fixed. 
This question on base points is part of a more general theory involving maps of spaces
$$\begin{matrix}
A & \xrightarrow{g} & B \\
i \downarrow && \downarrow j\\
X & \xrightarrow{f} & Y 
\end{matrix} $$
where $i,j$ are inclusions and $g,f$ are homotopy equivalences. Is then $(f,g)$ a homotopy equivalence of pairs? This is true if $i,j$ are cofibrations: see 7.4.2 of Topology and Groupoids.