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This is again related to the orientation. The following examples will clarify this point. In figure 3.7(a), an oriented 1-simplex $(p_0p_2)$ is divided into two, $(p_0p_1)$ and $(p_1p_2)$ . We agree that the boundary of $(p_0p_2)$ is $\\{p_0\\} \\cup \\{p_2\\}$ and so should be that of $(p_0p_1) + (p_1p_2)$ . If $\\partial_1(p_0p_2)$ were defined to be $p_0 + p_2$ , we would have $\\partial_1(p_0p_1) + \\partial_1(p_1p_2) = p_0 + p_1 + p_1 + p_2$ . This is not desirable since $p_1$ is a fictitious boundary. If, instead, we take $\\partial_1(p_0p_2) = p_2 - p_0$ , we will have $\\partial_1(p_0p_1) + \\partial_1(p_1p_2) = p_1 - p_0 + p_2 - p_1 = p_2 - p_0$ as expected. The next example is the triangle of figure 3.7(b). 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