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Nerve complex
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==Examples== # Let ''X'' be the circle <math>S^1</math> and <math>C = \{U_1, U_2\}</math>, where <math>U_1</math> is an arc covering the upper half of <math>S^1</math> and <math>U_2</math> is an arc covering its lower half, with some overlap at both sides (they must overlap at both sides in order to cover all of <math>S^1</math>). Then <math>N(C) = \{ \{1\}, \{2\}, \{1,2\} \}</math>, which is an abstract 1-simplex. # Let ''X'' be the circle <math>S^1</math> and <math>C = \{U_1, U_2, U_3\}</math>, where each <math>U_i</math> is an arc covering one third of <math>S^1</math>, with some overlap with the adjacent <math>U_i</math>. Then <math>N(C) = \{ \{1\}, \{2\}, \{3\}, \{1,2\}, \{2,3\}, \{3,1\} \}</math>. Note that {1,2,3} is not in <math>N(C)</math> since the common intersection of all three sets is empty; so <math>N(C)</math> is an unfilled triangle.
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