We say a based topological space $(X,x_0)$ is well-pointed (also known as non-degenerately based) if the inclusion $\{x_0\} \to X$ is not a cofibration (see May $\S$4.3).
For example, if $X$ is a CW complex based at a 0-cell, it is well-pointed. If $Y = \{0\}\cup\left\{\frac{1}{n} \ : \ n \in\mathbb{N}\right\}$ based at 0, then it is not well-pointed.
What are some other good examples of spaces that are not well-pointed? In particular what are examples of connected spaces which are not well-pointed?
The Hawaiian earring with basepoint at the origin is a connected pointed spaces that is not well-pointed.