Union of connected sets is connected

Question

I've been trying to solve two problems of connectedness without succeed.

1. Let $(X_\lambda)_\lambda \in L$ a family of connected sets, in a metric space M, such that $X_\lambda \cap X_\mu \neq \emptyset$ for every $\lambda, \mu \in L$. Prove that $\bigcup_{\lambda \in L} X_\lambda$ is connected.

I tried fixing $\lambda_0 \in L$, then $X_\lambda \cap X_{\lambda_0} \neq \emptyset$. Hence $X_\lambda \cup X_{\lambda_0}$ is connected for every $\lambda$. I don't know if I will conclude something useful with this argument.

2. Let $f:M\to N$ continuous and locally injective. If $M$ is connected and exists a continuous mapping $g:N\to M$ such that $f \circ g=id_N$, then $f$ is an homeomorphism of $M$ onto $N$.

Is sufficent to prove $g \circ f= id_M$ and I tried using the descomposition of $M=A \cup B$ where $A=\{x \in M: f \circ g(x)=x\}$ and $B=M\setminus A$. Clearly, A is closed, so if I can prove A is open, we are done. (well, if A is non empty).

Someone can help me?

Answer 1

Let $f\colon \bigcup X_\lambda\to\{0,1\}$ be a continuous function. Assume $f(x_1)=0$ and $f(x_2)=1$. Let $\lambda\in L$ with $x_1\in X_\lambda$. Then $f(x)=0$ for all $x\in X_\lambda$ because $X_\lambda$ is connected. Let $\mu\in L$ with $x_2\in X_\mu$. Then $f(x)=1$ for $x\in X_\mu$ by the same argument. For $x\in X_\lambda\cap X_\mu$, we arrive at a contradiction. Hence $f$ is constant. This shows that $\bigcup X_\lambda$ is connected.