Continuous partitions of unity

Let $X$ be a complete metric space, $M$ a closed set in it, $U_{1}, U_{2}, \ldots , U_{n}$ a cover of $M$ by open sets. Construct a system of continuous functions

called a partition of unity corresponding to the cover $U_{j}$, satisfying the conditions

1. $0 \leqslant e_{j}(x) \leqslant 1$ for every $x \in X$;

2. $e_{j}(x) = 0$, if $x \notin U_{j}$;

3. $\sum_{j = 1}^{n} e_{j}(x) = 1$ for every $x \in M$.

Formulate and prove an analogue of Theorem 10.8, in which $K$ is a compact subset of an arbitrary metric space. (Replace the functions $\varphi_{i}$ that occur in the proof of Theorem 10.8 by functions of the type constructed in Exercise 22 of Chap. 4).

Strengthen the conclusion of Theorem 10.8 by showing that the functions $\psi_{i}$ can be made differentiable, and even infinitely differentiable. (Use Exercise 1 of Chap. 8 in the construction of the auxiliary functions $\varphi_{i}$).