Remarks on the point character of Banach spaces and non-linear embeddings into c_0()

We give a brief survey of the results on coarse or uniform embeddings of Banach spaces into c_0() and the point character of Banach spaces. In the process we prove several new results in this direction (for example we determine the point character of the spaces L_p(μ), 1≤ p≤2) solving open problems posed by C. Avart, P. Komjath, and V. Roedl and by G. Godefroy, G. Lancien, and V. Zizler. In particular, we show that X=L_p(μ), 1≤ p<∞, bi-Lipschitz embeds into c_0() if and only if X<_.