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Seifert-Van Kampen theorem

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Revision as of 15:37, 14 May 2026 by InfernalAtom683 (talk | contribs) (Created page with "== Statement == Let <math>X</math> be a topological space, and <math>U, V\subset X</math> be open sets such that <math>X = U\cup V</math>, and <math>U</math>, <math>V</math> and <math>U\cap V</math> are path-connected. Take a basepoint <math>x_0\in U\cap V</math> with inclusion maps: <math display="block">i\colon U\cap V\hookrightarrow U,\quad j\colon U\cap V\hookrightarrow V,\quad k\colon U\hookrightarrow X,\quad l\colon V\hookrightarrow X,</math> then the following d...")

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Statement

Let $X$ be a topological space, and $U, V \subset X$ be open sets such that $X = U \cup V$ , and $U$ , $V$ and $U \cap V$ are path-connected. Take a basepoint $x_{0} \in U \cap V$ with inclusion maps:

$i : U \cap V ↪ U, j : U \cap V ↪ V, k : U ↪ X, l : V ↪ X,$

then the following diagram is a pushout:

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