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Summary: Math 446646
Important facts about Topological Spaces, Part II
Connected sets
· Let (X, T) be a topological space. Assume that there are two non-empty open
subsets A, B of X such that X = A B and A B = . Then {A, B} is a
partition of X.
· Notice that if {A, B} is a partition of X, the sets A and B are both open and
closed.
· A topological space (X, T) is connected if it does not admit any partition.
· Some characterizations of connectedness:
(a) (X, T) is connected if and only if the only subsets of X that are both open
and closed are X and .
(b) (X, T) is connected if and only if every continuous function f : X {0, 1}
is constant.
· Let (X, T) be a topological space, let {A}I be a collection of subsets of
X such that each (A, TA ) is connected. Assume that IA is non-empty.
Then IA is connected.
· Let (X, T) be a topological space, let A be a subset of X such that (A, TA) is
connected. Then any set B such that A B A is connected.
· Let (X, T) be connected and f : (X, T) (Y, T ) be continuous and onto.
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