Meaning of $C^0$ close between knots

Question

Can any one tell the actual meaning (geometrically too) of '$C^0$ close' in the following theorem?

"Given any topological knot $K$ there is a Legendrian knot $C^0$ close to it".

When I read "Jhon B Etnyre's, Legendrian and transversal knot" lecture notes I had this problem of understanding.

Thanks advance.

Answer 1

It means that for any $\epsilon >0$ and smooth knot $K$ (thought of as an embedding $K: S^1 \to S^3$), there is a Legendrian knot $L:S^1 \to S^3$ so that $d(K(x),L(x))< \epsilon$ for all $x$ where $d$ is the usual distance function on $S^3$.