  Comparing the module categories of an algebra and of the endomorphism algebra
of a given support $\tau$-tilting module, we give a generalization of the
Brenner-Butler's tilting theorem in the framework of $\tau$-tilting theory.
Afterwards we define $\tau$-slices and prove that complete slices of tilted
algebras and local slices of cluster tilted algebras are examples of complete
$\tau$-slices. Then we apply this concept to the study of simply connected
tilted algebras. Finally, we study the one-point extensions and the
split-by-nilpotent extensions of an algebra with $\tau$-slices.
