Let $X$ be a finite CW complex and $x_0$ a point in $X$.
My question is then just:
Is $X-\{x_0\}$ still homotopy equivalent to a finite CW complex?
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this communityLet $X$ be a finite CW complex and $x_0$ a point in $X$.
My question is then just:
Is $X-\{x_0\}$ still homotopy equivalent to a finite CW complex?
The answer to your question is no. Here is a counterexample.
Let $X$ be the CW-complex obtained by attaching a 2-cell to the space $[-1,1]$ via the attaching map $S^1\cong [-1,1]/(-1\sim 1) \longrightarrow [-1,1]$ given by (the continuous extension of) $x\mapsto x\sin(1/x)$.
Then $X\setminus \{0\}$ is homotopy equivalent to an infinite wedge of $S^1$'s.