- 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 and A \cup B = X. (Added! @Willie thanks!)
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:
- ...A and B whose union is Y and neither of which contains a limit point of other.
- A and B are both closed and open in Y and their union (in Y) is Y.