Connectedness via connected fibers of a quotient map

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Problem 2.1.15 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 2.1 "Point Set Topology"

(a)(b)

Let $q : X \to Y$ be a quotient space projection from a topological space $X$ to a connected topological space $Y$ . Assume that $q^{-1}(y)$ is connected for each $y \in Y$ .

(a)

Show that $X$ is connected.

(b)

Is $X$ necessarily connected if the map $q$ is not assumed to be a quotient mapping? Justify your assertion.

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.