  We explore the four-dimensional effective $F(T)$ gravity with $T$ the torsion
scalar in teleparallelism originating from higher-dimensional space-time
theories, in particular the Kaluza-Klein (KK) and Randall-Sundrum (RS)
theories. First, through the KK dimensional reduction from the five-dimensional
space-time, we obtain the four-dimensional effective theory of $F(T)$ gravity
with its coupling to a scalar field. Second, taking the RS type-II model in
which there exist the five-dimensional Anti-de Sitter (AdS) space-time with
four-dimensional Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) brane, we find
that there will appear the contribution of $F(T)$ gravity on the
four-dimensional FLRW brane. It is demonstrated that inflation and the dark
energy dominated stage can be realized in the KK and RS models, respectively,
due to the effect of only the torsion in teleparallelism without that of the
curvature.
