- #1
Carl140
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Homework Statement
Let (X,d) be a metric space, A subset of X, x_A: X->R the characteristic
function of A. (R is the set of all real numbers)
Let V_d(x) denote the set of neighbourhoods of x with respect the metric d.
Prove that x_A is continuous in x (x in X) if and only if there
exists an element V in V_d(x) such that V is a subset of A and V is a
subset of X\A.
The Attempt at a Solution
OK, so assume x_A is continuous then for each closed subset of R
the preimage of this closed subset under x_A must be closed.
Take V= {0} then V is closed so (x_A)^(-1) = { y in X: x_A(y) = 0} = X\A.
But I don't see how this helps..I don't know hwo to find such V in V_d(x).