Property Testing refers to the problem of making a small number of queries to determine whether $x$ is in some language $L$ or whether it is far away from being in $L$.
More precisely we want to distinguish
1) $x \in L$
2) $x$ epsilon-far from every $y \in L$.
Here, epsilon-far is generally taken to mean hamming distance.
I don't quite understand why property testing has this asymmetric definition, i.e. why don't we want 1) to be "$x$ epsilon-close to some $y \in L$"