In general, if you are given a subset of a topological space, which is itself referred to as a topological space, then it is assumed to have the subspace topology. Any subset of a topological space inherits the ambiant topology in a natural way. Let

be the inclusion map. The topology on

is the topology generated by the open sets of the form

, where

is open in

.

Now it is easy to solve the problem. Show that any open cover of a set in

lifts to an open cover of the same set, considered as a subset of

. Conversely, any open cover of some subset of

(considered as a subset of

) can be considered as an open cover of the same set as a set of

.

By the way, the notation

for the

-sphere is universal!