(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

3. 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).

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# Homework Help: Characteristic function

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