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?
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? 


The answer to your question is no. Here is a counterexample. Let $X$ be the CWcomplex obtained by attaching a 2cell 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. 

