Why is relative interior point not equivalent to interior point under the following definition?

Question

Let $C$ be a convex set an further define the relative interior point of $C$ as

$$\operatorname{ri}(C):=\{ x \in C: \forall y \in C, \exists \epsilon > 0, x-\epsilon(y-x) \in C\}$$

it is clear for me that for an interior point $x\in C$ that it is then also in $\operatorname{ri }(C)$

but I do not see why a relative interior point is not also an interior point? Does it have to do with the fact that I cannot necessarily find a constant radius to be contained in $C$? Any help is greatly appreciated.

Answer 1

For instance, let $C=[-1,1]\times \{0\}$ be a convex subset of $\Bbb R^2$ endowed with the standard topology. The set $C$ has empty interior in $\Bbb R^2$, but $(0,0)\in\operatorname{ri}(C)$. A less trivial example is a subset $C=\prod_{n=1}^{\infty} [-2^{-n};2^{-n}]$ of the space $\ell_2$. The set $C$ has empty interior in $\ell_2$, but $0\in\operatorname{ri}(C)$.