Interior point in convex set on n.v.s I'm reading Peter Lax's Functional analysis, and I have a question about a definition:


  Definition. $X$ is a linear space over the reals, $S$ a subset of $X$.  A point $x_0$ is called an interior point of $S$ if for any $y$ in $X$ there is an $\epsilon$, depending on $y$, such that
  $$x_0 + ty \in S \qquad\text{for all real $t$, $|t|&lt;\epsilon$.}$$


I'm wondering if this definition of interior point is equivalent to the classical one (topological interior) in the case of a convex set. I was looking for a counterexample but I couldn't find it.
It  is not the same thing. It is what is called the algebraic interior. The standard counterexample is below where the origin is not in the topological interior.
EDIT (again) : what follows is 100% wrong and it is a good example of how NOT to write an answer. 



EDIT: What's below is not true; it refers to the definition of "algebraic interior of a convex set", which is another thing. The definition in the original question is exactly the same as the usual one of interior point in the topological sense.



They're not equivalent. Consider a line segment in the plane. Its midpoint, or any point not in the extremes, is in the "algebraic interior" but not in the topological interior, since the latter is empty.