Interior point in convex set on n.v.s

Question

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 $$x_0 + ty \in S \qquad\text{for all real $t$, $|t|<\epsilon$.}$$

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.

Answer 1

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.

Answer 2

EDIT (again) : what follows is 100% wrong and it is a good example of how NOT to write an answer.

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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.

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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.