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.

  • 4
    $\begingroup$ Two points works $\endgroup$
    – Thomas Rot
    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.


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