Sea ${\displaystyle X}$  un espacio topológico, ${\displaystyle X=U_{1}\cup U_{2}}$ , con ${\displaystyle U_{1},U_{2}}$  subconjuntos abiertos y conexos por caminos, tales que ${\displaystyle U_{1}\cap U_{2}\neq \varnothing }$  también es conexo por caminos. Sea ${\displaystyle x_{0}\in U_{1}\cap U_{2}}$ .

Supongamos que conocemos los grupos fundamentales

${\displaystyle \pi _{1}(U_{1},x_{0})=\langle S_{1};R_{1}\rangle \,}$ ,

${\displaystyle \pi _{1}(U_{2},x_{0})=\langle S_{2};R_{2}\rangle \,}$  y

${\displaystyle \pi _{1}(U_{1}\cap U_{2},x_{0})=\langle S;R\rangle }$ .

Entonces, ${\displaystyle \pi _{1}(X,x_{0})=\langle S_{1}\cup S_{2};R_{1}\cup R_{2}\cup \{(i_{1})_{*}(s)((i_{2})_{*}(s))^{-1}|s\in S\}\rangle }$ , donde,
si ${\displaystyle i_{1}:U_{1}\cap U_{2}\rightarrow U_{1}}$  y ${\displaystyle i_{2}:U_{1}\cap U_{2}\rightarrow U_{2}}$  son las inclusiones naturales,
entonces ${\displaystyle (i_{1})_{*}}$  y ${\displaystyle (i_{2})_{*}}$  son las aplicaciones inducidas tales que

${\displaystyle (i_{1})_{*}:\pi _{1}(U_{1}\cap U_{2},x_{0})\rightarrow \pi _{1}(U_{1},x_{0})}$  que actúa ${\displaystyle [\alpha ]\rightarrow (i_{1})_{*}([\alpha ]):=[i_{1}\circ \alpha ]}$ ,

y análogamente

${\displaystyle (i_{2})_{*}:\pi _{1}(U_{1}\cap U_{2},x_{0})\rightarrow \pi _{1}(U_{2},x_{0})}$  que actúa ${\displaystyle [\alpha ]\rightarrow (i_{2})_{*}([\alpha ]):=[i_{2}\circ \alpha ]}$ .

• Grupoide

• Allen Hatcher, Algebraic topology. (2002) Cambridge University Press, Cambridge, xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
• Peter May, A Concise Course in Algebraic Topology. (1999) University of Chicago Press, ISBN 0-226-51183-9 (Section 2.7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids).

• Van Kampen's theorem result en PlanetMath.