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1. Let $p:X \to Y$ be a quotient map. Suppose that [tex]p^{-1}(y)[/tex] is connected for each [tex]y \in Y[/tex]. Show [tex]X[/tex] is connected if an only if [tex]Y[/tex] is connected.

2. Let [tex](X,\mathcal{T})[/tex] be a topological space and $A,B \subseteq X$. Suppose $A$ is open, $\mbox{cl}(A) \subseteq B$, $B$ is connected and $\partial A$ is connected. Proof $B\setminus A$ is connected.

The problem is I don't know a good idea how to prove a set is connected. For the first question, it's clear that $Y$ is connected when $X$ is connected because $p$ is a quotient map. But I don't know how to prove the other implication.

I have no clue for the second question.

Anyone?