In my version of Bott and Tu, the example reads (emphasis mine):

Moreover, if $V \subset U$ is an inclusion of CONTRACTIBLE open sets, then $\rho^U_V: H^q(\pi^{-1} U) \to H^q( \pi^{-1} V)$ is an isomorphism.

Looks like this is the original edition, copyright 1982, 4th printing.

In my version of Bott and Tu, the example reads (emphasis mine):

"Moreover, if $V \subset U$ is an inclusion of CONTRACTIBLE open sets, then $\rho^U_V: H^q(\pi^{-1} U) \to H^q( \pi^{-1} V)$ is an isomorphism."

It goes on to say that this "is an example of a locally constant presheaf on a good cover, to be defined in Section 13." The idea then is that in this context one only allows certain open sets as inputs.

Looks like this is the original edition, copyright 1982, 4th printing.

In my version of Bott and Tu, the example reads (emphasis mine):

Moreover, if $V \subset U$ is an inclusion of CONTRACTIBLE open sets, then $\rho^U_V: H^q(\pi^{-1} U) \to H^q( \pi^{-1} V)$ is an isomorphism.

In my version of Bott and Tu, the example reads (emphasis mine):

Moreover, if $V \subset U$ is an inclusion of CONTRACTIBLE open sets, then $\rho^U_V: H^q(\pi^{-1} U) \to H^q( \pi^{-1} V)$ is an isomorphism.

Looks like this is the original edition, copyright 1982, 4th printing.