# A space $X$ where $X$ minus a point is homotopically trivial, but $X$ isn't

An example of a space $$X$$ that is not homotopy equivalent to a point, but there exists a $$p \in X$$ such that $$X \backslash \{p\}$$ is homotopy equivalent to a point.

• Two points works Apr 18 '20 at 18:11

Hint: How about $$S^1$$? Not "homotopy equivalent to a point" you have via the fundamental group $$\pi_1(S^1) = \mathbb{Z}$$ for example, so now what about the complement of a point? At least intuitively it should be clear why this example works.