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?
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6$\begingroup$ The answer is yes if $x_0$ lies in the interior of a top dimensional cell. $\endgroup$– John KleinCommented Jun 11, 2012 at 13:41
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8$\begingroup$ The answer is also yes if $X$ is a regular CW-complex (i.e. if all its attaching maps are homemorphisms). $\endgroup$– André HenriquesCommented Jun 11, 2012 at 14:38
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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.
answered Jun 11, 2012 at 13:38
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1$\begingroup$ All I have to say about this example is "yuck". Good answer though. $\endgroup$ Commented Jun 11, 2012 at 15:30
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15$\begingroup$ There's nothing yuck about this particular example. But I'll agree that the notion of CW-complex is a bit "yuck". Depending on one's perspective, the notion of CW-complex is either too general, or not general enough. The two notions that I believe to have a "good level of generality" are 1) regular CW-complex, and 2) retract of a CW-complex. $\endgroup$ Commented Jun 11, 2012 at 15:42
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1$\begingroup$ @Matin: I guess that Andr\'e thinks of $\mathrm{S}^1$ as the quotient of the interval $[-1,1]$ by its boundary. Its attaching map takes the same value at $-1$ and at $1$. $\endgroup$– OblomovCommented Jun 12, 2012 at 8:11
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1$\begingroup$ @Andr\'e: Actually, are you sure that the resulting space is really a a infinite wedge of circles. It seems to me that it might even not be a CW complex but rather a ''sort of Hawaian earrings`` space. $\endgroup$– OblomovCommented Jun 12, 2012 at 9:02
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7$\begingroup$ The Hawaiian earing space has a sequence of loops that converge to a constant loop. In the space I constructed, there is no sequence of loops converging to a constant loop, for the simple reason that the limit point has been removed. Here's another example where a similar phenomenon shows up: $\mathbb R^2\setminus \{1/2,1/3,1/4,...\}$ is h.e. to the Hawaiian earings, but $\mathbb R^2\setminus (\{0\}\cup\{1/2,1/3,1/4,...\})$ is homotopy equivalent to an infinite wedge of $S^1$s. $\endgroup$ Commented Jun 12, 2012 at 9:07