Exercise 2, § 2

Let $Y$ be a uniform space and let $X$ be a subspace of $Y$. Let $(U_i)_{i\in I}$ be an open covering of $X$.

a) Prove that there exists an open neighbourhood V of X in Y and an open covering $(V_i)_{i\in I}$ of V satisfying the following properties:

(i) For every $i,U_i= V_i\cap X$ ;

(ii) For every subset J of I such that $\bigcap_{j\in J}V_j=\not\emptyset ,\bigcap_{j\in J}U_j=\not\emptyset$.

b) Suppose that Y is locally connected and that the $U_i$ are connected. Prove that there exists such a covering where the $V_i$ are connected open subsets of Y.

c) Suppose that the $U_i$ are connected and let, for every $i\in I$, a point $x_i\in U_i$. Suppose that the image of $\pi_1(U_i, x_i)$ in $\pi_1(X, x_i)$ is reduced to the identity element. Now prove that there exists a neighbourhood V of X in Y such that, for every $x\in X$, the canonical homomorphism $\pi_1(X, x)\rightarrow \pi_1(V, x)$ admits a retraction.