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Slats18
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Homework Statement
Let (A,S) and (B,T) be topological spaces and let f : A -> B be a continuous function. Suppose that D is dense in A, and that (B,T) is a Hausdorff space. Show that if f is constant on D, then f is constant on A.
Homework Equations
D is a dense subset of (A,S) iff the intersection of D and U is not the empty set, for every non-empty open set U.
D is dense on A, i.e., the closure of D is A.
(B,T) is Hausdorff, i.e., for any two points in B, there exist disjoint open sets U,V in T such that the intersection of U and V is the empty set.
The Attempt at a Solution
Suppose, by way of contradiction, that this is not true. Then, f is not constant on A, if f is constant on D.
I'm not sure how I go about this, I can't assume f(D) is dense in B because we haven't been told it's onto, only continuous, correct? How can I show f is constant on D?