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Aug 16, 2013 at 23:07 comment added Eric Wofsey The way I imagine proving the "not hard to see" part is to show (eg, by induction on the number of cells in $K$) that if $K$ contains the singular point of $X$, $K$ is contained in a contractible subset whose boundary is a union of two loops contained in the 1-skeleton of $X$. It is quite possible that you can do away with the finiteness assumption with a little more work.
Aug 16, 2013 at 23:00 comment added Vidit Nanda Where do you use the finiteness of $K$ in the part that "is not hard to see"?
Aug 16, 2013 at 21:47 history answered Eric Wofsey CC BY-SA 3.0