- A subset of a topological space is naturally endowed with a topology, namely, the subspace topology. 

NOTE: Whenever we speak of a topology on a subspace, unless specified otherwise, we mean this subspace topology. It seems natural to assume the following definition of a connected topological space (and Munkres does so) :

 -  A topological space X is connected if for any two nonempty open sets A and B of X, A \cap B is nonempty. 

It follows that any subspace of X is connected if it is connected with respect to the induced (subspace) topology on it.

 -  A connected subspace is a subset which is a connected space wrt the induced topology. (A connected component is a maximal (wrt to inclusion) connected subset of X. )

Now, Munkres gives a characterization of connectedness of a subspace:

> If Y is subspace of X, a separation of
> Y is a pair of disjoint nonempty sets
> A and B whose union is Y, neither of
> which contains a limit point of the
> other. The space Y is connected if
> there exists no separation of Y.

The proof given there is clear. One point to note is that the following are equivalent for subsets A and B **of X**:

1. ...A and B whose union is Y and neither of which contains a limit point of other.
2. A and B are both closed and open **in Y** and their union (in Y) is Y.