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• 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 whenever any two nonempty open sets A and B of X cover X, then A \cap B is nonemptyand 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:

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.
2 added 42 characters in body
• 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:

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.
1
• 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.