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In Cellular Boundary Formula, one is asked to calculate the degree of a map $S^{n-1}\rightarrow X_{n-1}/(X_{n-1}\backslash e_{n-1}^\beta)\rightarrow S^{n-1}$, where the first sphere denotes the boundary of the n-cell $e_{n}^\alpha$, the second sphere was made up from an (n-1) cell by collapsing all $e_{n-1}^\beta$'s neighbors into a point.

So my question is: When n=0, how can I make a 0-sphere from a 0-cell which is only a point?

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One generally defines $S^{-1}=\emptyset$ as the empty set. This makes sense since $D^0=\ast$ is the $1$-point space so has empty boundary. You have for any space $X$ that $X/\emptyset=X_+$ is the disjoint union of $X$ with a disjoint basepoint and from this you get $D^0/\emptyset=\ast/\emptyset=S^0$.

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