# Homomorphisms induced by inclusions

Let $X$ be a topological space and $U, V ⊂ X$ open sets such that $X = U ∪ V$; $U ∩ V$ is path-connected; $V$ is simply connected. Let $x_0 ∈ U ∩ V$.

• Is the homomorphism $i_* : π_1 (U, x_0)\to π_1 (X, x_0)$, induced by the inclusion, surjective?
• Is it true that, for a subspace $W ⊂ X$ (not necessarily open), if $U ⊂ W$, then the homomorphism $j_* : π_1 (W, x_0)\to π_1 (X, x_0)$, induced by the inclusion, is surjective?

Thanks!

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use van kampen. –  gmoss Feb 11 '13 at 3:41