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Definition 1.1. A topological space (X, T) is disconnected if and only if there exist disjoint, non-empty open subsets A and B of X such that X = A cup B. Otherwise, we say that (X, T) is connected. As the next exercise illustrates, connectedness relies on the topology on X, not the set itself. Exercise 1.2. (HW) Show that (mathbb{R}) with the discrete topology is disconnected and that (mathbb{R}) with the trivial topology is connected. Proposition 1.3. Let (X, T) be a topological space. Then X is disconnected if and only if there are non-empty, disjoint closed subsets A, B of X such that X = A cup B. Proposition 1.4. A topological space (X, T) is connected if and only if the only subsets of X that are both open and closed are (emptyset) and X. Connectedness of a subspace of a topological space has a similar description. Proposition 1.5. Let Y be a subspace of a topological space (X, T). The following are equivalent: (1) Y is disconnected. (2) There exist open subsets A, B of X such that (Y subseteq A cup B), (A cap Y eq emptyset), (B cap Y eq emptyset), and (A cap B subseteq X - Y). (3) There exist closed subsets A, B of X such that (Y subseteq A cup B), (A cap Y eq emptyset), (B cap Y eq emptyset), and (A cap B subseteq X - Y).

          Definition 1.1. A topological space (X, T) is disconnected if and only if there
exist disjoint, non-empty open subsets A and B of X such that X = A cup B.
Otherwise, we say that (X, T) is connected.
As the next exercise illustrates, connectedness relies on the topology on X, not
the set itself.
Exercise 1.2. (HW) Show that (mathbb{R}) with the discrete topology is disconnected and
that (mathbb{R}) with the trivial topology is connected.
Proposition 1.3. Let (X, T) be a topological space. Then X is disconnected if and
only if there are non-empty, disjoint closed subsets A, B of X such that X = A cup B.
Proposition 1.4. A topological space (X, T) is connected if and only if the only
subsets of X that are both open and closed are (emptyset) and X.
Connectedness of a subspace of a topological space has a similar description.
Proposition 1.5. Let Y be a subspace of a topological space (X, T). The following
are equivalent:
(1) Y is disconnected.
(2) There exist open subsets A, B of X such that (Y subseteq A cup B), (A cap Y 
eq emptyset),
(B cap Y 
eq emptyset), and (A cap B subseteq X - Y).
(3) There exist closed subsets A, B of X such that (Y subseteq A cup B), (A cap Y 
eq emptyset),
(B cap Y 
eq emptyset), and (A cap B subseteq X - Y).

Definition 1.1. A topological space (X, T) is disconnected if and only if there
exist disjoint, non-empty open subsets A and B of X such that X = A cup B.
Otherwise, we say that (X, T) is connected.
As the next exercise illustrates, connectedness relies on the topology on X, not
the set itself.
Exercise 1.2. (HW) Show that (mathbbR) with the discrete topology is disconnected and
that (mathbbR) with the trivial topology is connected.
Proposition 1.3. Let (X, T) be a topological space. Then X is disconnected if and
only if there are non-empty, disjoint closed subsets A, B of X such that X = A cup B.
Proposition 1.4. A topological space (X, T) is connected if and only if the only
subsets of X that are both open and closed are (emptyset) and X.
Connectedness of a subspace of a topological space has a similar description.
Proposition 1.5. Let Y be a subspace of a topological space (X, T). The following
are equivalent:
(1) Y is disconnected.
(2) There exist open subsets A, B of X such that (Y subseteq A cup B), (A cap Y
eq emptyset),
(B cap Y
eq emptyset), and (A cap B subseteq X - Y).
(3) There exist closed subsets A, B of X such that (Y subseteq A cup B), (A cap Y
eq emptyset),
(B cap Y
eq emptyset), and (A cap B subseteq X - Y).

Added by Brooke C.

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