Homotopy lifting property for covering maps

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Problem 4.1.20 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 4.1 "Basic examples and properties"

(a)(b)

Prove the following theorem: Suppose that $F : (I \times I, \left\{ 0\right\} \times I, 1 \times I) \to (B, b_1, b_2)$ is a homotopy between $f_0, f_1$ , and $p(a) = b_1$ .

(a)

There is a unique lifting $\tilde{F}$ with $\tilde{F}(0,0) = a$ .

(b)

If $\tilde{f}_0, \tilde{f}_1$ are liftings with $\tilde{f}_0(0) = \tilde{f}_1(0) = a$ , then $\tilde{f}_0(1) = \tilde{f}_1(1)$ .

References

[Law06] Lawson T. Topology: A Geometric Approach. Oxford University Press, 2006.