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$$ (X,T)\textrm{ is a topoligical space. }U\in T.\, \, \, T^{'}=\left \{ U\cap V\mid V\in T \right \}\\ x_{0}\in U. \, \,\, \, i:U\rightarrow X\textrm{ is the inclusion map }.\\ \textrm{Is the induced homomorphism }i_{*}:\pi (U,x_{0})\rightarrow \pi (X,x_{0})\textrm{ injective ?}$$

This is a doubt that I have while trying to prove a special case of Seifert-van Kampen theorem from Munkres' Topology.

I edited this question to assume U is open in X. Earlier I asked for arbitrary subset U of X.

tony
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    No, there's no reason for this to be true. Consider, for example, the circle $S^1 \subsetneq \mathbb{R}^2$. The problem is that paths which are not homotopic in $U$ may become homotopic in $X$. – Qiaochu Yuan Oct 26 '20 at 03:27
  • What if U is open – tony Oct 26 '20 at 03:28
  • Then you can thicken the circle to the open annulus, there's still no reason for this to be true. – Qiaochu Yuan Oct 26 '20 at 03:30
  • If this were true even just for $U$ open then every open subset of a simply connected space would be simply connected, which is clearly false. – Matematleta Oct 26 '20 at 03:37

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