In my text, it says
"Given a topological space $X$ and a subspace $S ⊂ X$, define the induced topology on $S$ to be the topology in which the open sets are of form $U ∩ S$, where $U$ is open in $X$ and $S^n$ (the n-sphere) with its induced topology is a manifold"
Can someone rephrase this or clarify what it means for a topology (a collection of open sets) to be an induced topology?
Look at an example. The topology of $[0,1]\subset \mathbb{R}$ has as open sets the intersections of open sets of $\mathbb{R}$ and $[0,1]$. So for instance $(1/2,3/4)$, $(3/4,1]$ and $[0,1)$ are open sets of $[0,1]$ in the topology induced by the topology if the real line.
It's also often called the subspace topology: http://en.wikipedia.org/wiki/Subspace_topology
It provides a subset of a topological space with a topology of its own, and it works in the way you might expect. The subspace topology on $S\subseteq X$ is the one in which a subset of $S$ is open iff it is the intersection of an open subset of $X$ with $S$.
In your example you're considering n-sphere as a subset of $(n+1)$-dimensional euclidean space. Equip $S^n$ with the subspace (induced) topology. The result should be a topological manifold.
Induced basically means it is generated by another through some mean. In this case the subspace $S$ has a topology being generated by the topology of $X$ through the process of intersection.
They are referring to the subspace topology on the subset $S \subseteq X$. Others have already explained this in terms of intersections of $X$-open sets with $S$, but you can also think of it in terms of the characteristic property of the topology: the subspace topology on $S$ is the coarsest topology such that the inclusion $S \hookrightarrow X$ is continuous. This means that a function $g: Z \to S$ is continuous if and only if the composite $Z \to S \hookrightarrow X$ is continuous, so it tells you everything you need to know about what maps into $S$ are continuous. It is a specific example of an initial topology, which you will also see when you discuss products of topological spaces (and pullbacks), so it helps to get familiar with it early on.
